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%  T is Tarski's B,  non-strict betweenness
%  E is equidistance
%  Names for the axioms as in SST.
% Proves the third and last case of Lemma 12.11A.
% This lemma is the first claim of the proof of Satz 12.11, which
% is insufficiently justified in SST.  Namely, there exists a point c' 
% (here c2) of C,  on the opposite side of B from t.
% A correct argument goes like this:
%  Pick c2!= a on line C.   
%  If c2 is on the opposite side of B from t, then pick b to lie between 
% them on B and pick c1=c2.   Otherwise c2 is on the same side of B from t.  
% This is the hard part;  apparently SST completely overlooked the difficulty.
% Then let c1 = s(a,c2);  then c1 is on the opposite side of B from c2 and 
% hence the opposite side from t.  Then pick b on B between c1 and t.  
% This proof is far more difficult than "da B und C sich in a schneiden.",
% which is no justification at all.  We use Lemma 9.13f for this, which 
% is proved only using the dimension 2 axiom, so we don't fix Szmielew's
% proof for n dimensions here.
 


set(hyper_res).
clear(order_hyper).
set(para_into).
set(para_from).
set(binary_res).
%set(neg_hyper_res).
set(ur_res).
set(para_from).
set(para_into).
%  set(unit_deletion).
set(order_history).
assign(report,5400).
%  assign(max_seconds, 36000).
assign(max_mem,840000).
%clear(print_kept).
%set(very_verbose).
set(input_sos_first).
%  set(ancestor_subsume).
set(back_sub).
% set(sos_queue).



assign(max_weight,15).
assign(max_distinct_vars,3).
assign(pick_given_ratio,2).
assign(max_proofs,1).
assign(heat,2).

assign(bsub_hint_wt,-1).
%assign(fsub_hint_wt,-1).
set(keep_hint_subsumers).


weight_list(pick_and_purge).
weight(alpha,4).
weight(beta,4).
weight(gamma,4).
end_of_list.

list(hot).    
end_of_list.

list(demodulators).
 il(t,c,r,cs)=b2.
  ext(p1,p2,p1,a)=c2.   % Then c2 is on line C and not equal to a. 
 s(a,c2)=c.
il(t,c2,r,cs) = b.
end_of_list.
set(lrpo).




list(usable).    

x=x.
E(x,y,y,x).                                   % A1 from page 10 of sst
-E(x,y,z,v) | -E(x,y,z2,v2) | E(z,v,z2,v2).   % A2
-E(x,y,z,z) | x=y.                            % A3
T(x,y,ext(x,y,w,v)).                          % A4, first half
E(y,ext(x,y,w,v),w,v).                        % A4, second half
-E(x,y,x1,y1) | -E(y,z,y1,z1) | -E(x,v,x1,v1) | -E(y,v,y1,v1) |
-T(x,y,z) | -T(x1,y1,z1) | x=y | E(z,v,z1,v1).  % A5
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | xa = xb | E(xc,xd,xc1,xd1).
% A5 equivalent form
% Following defines AFS
-T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xb,xa1,xb1)
| -E(xb,xc,xb1,xc1) | -E(xa,xd,xa1,xd1) | -E(xb,xd,xb1,xd1)
| AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1).
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | T(xa,xb,xc).
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | T(xa1,xb1,xc1).
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xa,xb,xa1,xb1).
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xa,xd,xa1,xd1).
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xc,xb1,xc1).
-AFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1).
-T(x,y,x) | x=y.                              % A6
% A7, inner Pasch, two clauses.
-T(xa,xp,xc) | -T(xb,xq,xc) | T(xp,ip(xa,xp,xc,xb,xq),xb).
-T(xa,xp,xc) | -T(xb,xq,xc) | T(xq,ip(xa,xp,xc,xb,xq),xa).
-T(alpha,beta,gamma).  %A8,  three lines.
-T(beta,gamma,alpha).
-T(gamma,alpha,beta).

E(x,y,x,y).  % Satz2-1
-E(xa,xb,xc,xd) | E(xc,xd,xa,xb).  % Satz2-2
-E(xa,xb,xc,xd) | E(xb,xa,xc,xd).  % Satz2-4
-E(xa,xb,xc,xd) | -E(xc,xd,xe,xf) | E(xa,xb,xe,xf).  %Satz2-3
-E(xa,xb,xc,xd) | E(xa,xb,xd,xc).  % Satz2-5
E(x,x,y,y).  % Satz 2-8
-T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xb,xa1,xb1) |
-E(xb,xc,xb1,xc1) | E(xa,xc,xa1,xc1).  % Satz 2.11
xq = xa | -T(xq,xa,u) | -E(xa,u,xc,xd) | ext(xq,xa,xc,xd) = u.    % Satz 2.12
T(x,y,y).    % Satz 3.1
-T(xa,xb,xc) | T(xc,xb,xa).  % Satz 3.2.
T(xa,xa,xb).  % Satz 3.3
-T(xa,xb,xc) | -T(xb,xa,xc) | xa = xb.  % Satz 3.4.
-T(xa,xb,xd) | -T(xb,xc,xd) | T(xa,xb,xc). % Satz 3.51.
-T(xa,xb,xd) | -T(xb,xc,xd) | T(xa,xc,xd). % Satz 3.52.
-T(xa,xb,xc) | -T(xa,xc,xd) | T(xb,xc,xd). % Satz 3.61.
-T(xa,xb,xc) | -T(xa,xc,xd) | T(xa,xb,xd). % Satz 3.61.
-T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xc,xd).  % Satz 3.71
-T(xa,xb,xc) | -T(xa,xc,xd) | T(xa,xb,xd). % Satz 3.62.
-T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xc,xd).  % Satz 3.71
-T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xb,xd).  % Satz 3.72
-IFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1).  % Satz 4.2 
% Satz 4.2  expressed directly by expanding IFS:   
-T(xa,xb,xc) | -T(za,zb,zc) | -E(xa,xc,za,zc) | -E(xb,xc,zb,zc)
| -E(xa,xd,za,zd) | -E(xc,xd,zc,zd)   | E(xb,xd,zb,zd).  
-T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xc,xa1,xc1)
| -E(xb,xc,xb1,xc1) | E(xa,xb,xa1,xb1).  % Satz 4.3

alpha != beta.  % Satz 3.13
beta != gamma.
alpha != gamma.
T(xa,xb,ext(xa,xb,alpha,gamma)).   % Satz 3.14, first half
xb != ext(xa,xb,alpha,gamma).    % Satz 3.14, second half
% The following many clauses are Definition 4.1
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | T(xa,xb,xc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | T(za,zb,zc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xc,za,zc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xb,xc,zb,zc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xd,za,zd).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xc,xd,zc,zd).
-T(xa,xb,xc) | -T(za,zb,zc) | -E(xa,xc,za,zc) | -E(xb,xc,zb,zc)
| -E(xa,xd,za,zd) | -E(xc,xd,zc,zd) | IFS(xa,xb,xc,xd,za,zb,zc,zd).

% Following 4 are definition 4.4 for n=3
-E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa1,xa2,xb1,xb2).
-E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa1,xa3,xb1,xb3).
-E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa2,xa3,xb2,xb3).
-E(xa1,xa2,xb1,xb2) | -E(xa1,xa3,xb1,xb3) | -E(xa2,xa3,xb2,xb3)
| E3(xa1,xa2,xa3,xb1,xb2,xb3).
% Following three lines are Satz 4.5
-T(xa,xb,xc) | -E(xa,xc,xa1,xc1) | T(xa1,insert(xa,xb,xa1,xc1),xc1).
-T(xa,xb,xc) | -E(xa,xc,xa1,xc1) | E3(xa,xb,xc,xa1,insert(xa,xb,xa1,xc1),xc1).
insert(xa,xb,xa1,xc1) = ext(ext(xc1,xa1,alpha,gamma),xa1,xa,xb).
-T(xa,xb,xc) | -E3(xa,xb,xc,xa1,xb1,xc1) | T(xa1,xb1,xc1).  % Satz 4.6

% following is Definition 4.10
-Col(xa,xb,xc) | T(xa,xb,xc) | T(xb,xc,xa) | T(xc,xa,xb).
Col(xa,xb,xc) | -T(xa,xb,xc).
Col(xa,xb,xc) | -T(xb,xc,xa).
Col(xa,xb,xc) | -T(xc,xa,xb).
% Following are Satz 4.11
-Col(x,y,z) | Col(y,z,x).
-Col(x,y,z) | Col(z,x,y).
-Col(x,y,z) | Col(z,y,x).
-Col(x,y,z) | Col(y,x,z).
-Col(x,y,z) | Col(x,z,y).
% following is Satz 4.12
Col(x,x,y).
% following is Satz 4.13
-Col(xa,xb,xc) | - E3(xa,xb,xc,xa1,xb1,xc1) | Col(xa1,xb1,xc1).
% following is Satz 4.14
-Col(xa,xb,xc) | -E(xa,xb,xa1,xb1)
| E3(xa,xb,xc,xa1,xb1,insert5(xa,xb,xc,xa1,xb1)).
% following is Definition 4.15
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | Col(xa,xb,xc).
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E3(xa,xb,xc,xa1,xb1,xc1).
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xa,xd,xa1,xd1).
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1).
-Col(xa,xb,xc) | - E3(xa,xb,xc,xa1,xb1,xc1) | - E(xa,xd,xa1,xd1)
| -E(xb,xd,xb1,xd1) | FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1).
% Following is Satz 4.16
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | xa = xb | E(xc,xd,xc1,xd1).
% Following is Satz 4.17
xa = xb | -Col(xa,xb,xc) | -E(xa,xp,xa,xq) | -E(xb,xp,xb,xq) | E(xc,xp,xc,xq).
% Following is Satz 4.18
xa = xb | -Col(xa,xb,xc) | -E(xa,xc,xa,xc1) | -E(xb,xc,xb,xc1) | xc = xc1.
% Following is Satz 4.19
-T(xa,xc,xb) | -E(xa,xc,xa,xc1) | -E(xb,xc,xb,xc1) | xc = xc1.
% Following is Satz 5.1
xa = xb | -T(xa,xb,xc) | -T(xa,xb,xd) | T(xa,xc,xd) | T(xa,xd,xc).
% Following is Satz 5.2
xa = xb | -T(xa,xb,xc) | -T(xa,xb,xd)| T(xb,xc,xd) | T(xb,xd,xc).
% Following is Satz 5.3
-T(xa,xb,xd) | -T(xa,xc,xd) | T(xa,xb,xc) | T(xa,xc,xb).
% Following is Definition 5.4
-T(xc,y,xd) | -E(xa,xb,xc,y) | le(xa,xb,xc,xd).
-le(xa,xb,xc,xd) | T(xc,insert(xa,xb,xc,xd),xd). %ab inserted into cd
-le(xa,xb,xc,xd) | E(xa,xb,xc,insert(xa,xb,xc,xd)).
% Following is Satz 5.5a
-le(xa,xb,xc,xd) | T(xa,xb,insert(xc,xd,xa,xb)).
-le(xa,xb,xc,xd) | E(xa,insert(xc,xd,xa,xb),xc,xd).
% Following is Satz 5.5b
-T(xa,xb,x) | -E(xa,x,xc,xd) | le(xa,xb,xc,xd).
% Following is Satz 5.6
-le(xa,xb,xc,xd) | -E(xa,xb,xa1,xb1) | -E(xc,xd,xc1,xd1) | le(xa1,xb1,xc1,xd1).
% Following is Satz 5.7
le(xa,xb,xa,xb).
% Following is Satz 5.8
-le(xa,xb,xc,xd) | - le(xc,xd,xe,xf) | le(xa,xb,xe,xf).
% Following is Satz 5.9
-le(xa,xb,xc,xd) | -le(xc,xd,xa,xb) | E(xa,xb,xc,xd).
% Following is Satz 5.10
le(xa,xb,xc,xd) | le(xc,xd,xa,xb).
% Following is Satz 5.11
le(xa,xa,xc,xd).
% Following is Definition 6.1
sameside(xa,xp,xb) | xa=xp | xb = xp | -T(xp,xa,xb).
sameside(xa,xp,xb) | xa=xp | xb = xp | -T(xp,xb,xa).
-sameside(xa,xp,xb) | xa != xp.
-sameside(xa,xp,xb) | xb != xp.
-sameside(xa,xp,xb) | T(xp,xa,xb) | T(xp,xb,xa).
%Following is Satz 6.2
xa=xp | xb=xp | xc = xp | -T(xa,xp,xc) | -T(xb,xp,xc) | sameside(xa,xp,xb).
xa=xp | xb=xp | xc = xp | -T(xa,xp,xc) | T(xb,xp,xc) | -sameside(xa,xp,xb).
% following is Satz 6.3
-sameside(xa,xp,xb) | xa != xp.
-sameside(xa,xp,xb) | xb != xp.
-sameside(xa,xp,xb) | f63(xa,xp,xb) != xp.
-sameside(xa,xp,xb) | T(xa,xp,f63(xa,xp,xc)).
-sameside(xa,xp,xb) | T(xb,xp,f63(xa,xp,xc)).
xa=xp | xb=xp | xc=xp | -T(xa,xp,xc) | -T(xb,xp,xc) | sameside(xa,xp,xb).
% following is Satz 6.4
-sameside(xa,xp,xb) | Col(xa,xp,xb).
-sameside(xa,xp,xb) | -T(xa,xp,xb).
-Col(xa,xp,xb) | T(xa,xp,xb) | sameside(xa,xp,xb).
% following is Satz 6.5
xa=xp | sameside(xa,xp,xa).
% Following is Satz 6.6
-sameside(xa,xp,xb) | sameside(xb,xp,xa).
% Following is Satz 6.7
-sameside(xa,xp,xb) | -sameside(xb,xp,xc) | sameside(xa,xp,xc).
% Following is Satz 6.11
xr = xa | xb = xc | sameside(insert(xb,xc,xa,xr),xa,xr).
xr = xa | xb = xc | E(xa,insert(xb,xc,xa,xr),xb,xc).
xr = xa | xb = xc | -sameside(x,xa,xr) | -sameside(y,xa,xr) |
-E(xa,x,xb,xc) | -E(xa,y,xb,xc) | x=y.
% Following is Satz 6.13
-sameside(xa,xp,xb) | -le(xp,xa,xp,xb) | T(xp,xa,xb).
-sameside(xa,xp,xb) | -T(xp,xa,xb) | le(xp,xa,xp,xb).
% Following is Satz 6.15
xp = xq | xp = xr | -T(xq,xp,xr) | -Col(xa,xp,xq) | xa = xp |
sameside(xa,xp,xq) | sameside(xa,xp,xr).
xp = xq | xp = xr | -T(xq,xp,xr) | -sameside(xa,xp,xq) | Col(xa,xp,xq).
xp = xq | xp = xr | -T(xq,xp,xr) | -sameside(xa,xp,xr) | Col(xa,xp,xq).
xp = xq | xp = xr | -T(xq,xp,xr) | xa != xp | Col(xa,xp,xq).

% Following is Definition 7.1, m is the midpoint of ab
-M(xa,xm,xb) | T(xa,xm,xb).
-M(xa,xm,xb) | E(xm,xa,xm,xb).
-T(xa,xm,xb) | -E(xm,xa,xm,xb) | M(xa,xm,xb).

-M(xa,xm,xb) | M(xb,xm,xa).   % Satz 7.2
M(xa,xa,xa).    % Satz 7.3a
-M(xa,xm,xa) | xm = xa.   % Satz 7.3b
-M(x,xa,z) | -M(x,xa,y) | z=y.  %Satz 7.4a
M(x,xa,s(xa,x)).     % Satz 7.4a,  and definition 7.5
-M(x,y,z) | z = s(y,x).   % Satz 7.6
s(x,s(x,y)) = y.    % Satz 7.7
s(xa,xp) != xr | s(xa,xq) != xr | xp = xq.   %Satz 7.8
s(xa,xp) != s(xa,xq) | xp = xq.    % Satz 7.9
s(xa,xp) != xp | xp = xa.   %Satz 7.10a
xp != xa | s(xa,xp)=xp.     % Satz 7.10b
E(xp,xq,s(xa,xp),s(xa,xq)).   % Satz 7.13
-T(xp,xq,xr) | T(s(xa,xp),s(xa,xq),s(xa,xr)).   % Satz 7.15a
T(xp,xq,xr) | -T(s(xa,xp),s(xa,xq),s(xa,xr)).   % Satz 7.15b
-E(xp,xq,xr,xs) | E(s(xa,xp),s(xa,xq),s(xa,xr),s(xa,xs)).  % Satz 7.16a
E(xp,xq,xr,xs) | -E(s(xa,xp),s(xa,xq),s(xa,xr),s(xa,xs)).  % Satz 7.16b
-M(xp,xa,xq) | -M(xp,xb,xq) | xa = xb.   % Satz 7.17
s(xa,xp) != s(xb,xp) | xa = xb.    % Satz 7.18
s(xa,s(xb,xp)) != s(xb,s(xa,xp)) | xa = xb.  % Satz 7.19a
s(xa,s(xb,xp)) = s(xb,s(xa,xp)) | xa != xb.  % Satz 7.19b
-Col(xa,xm,xb) | -E(xm,xa,xm,xb) | xa = xb | M(xa,xm,xb).  % Satz 7.20
% Following 2 lines are Lemma 7.21
Col(xa,xb,xc) | xb = xd | -E(xa,xb,xc,xd) | -E(xb,xc,xd,xa) |
-Col(xa,xp,xc) | -Col(xb,xp,xd) | M(xa,xp,xc).
Col(xa,xb,xc) | xb = xd | -E(xa,xb,xc,xd) | -E(xb,xc,xd,xa) |
-Col(xa,xp,xc) | -Col(xb,xp,xd) | M(xb,xp,xd).
% following is 7.22,  Krippenlemma
-T(xa1,xc,xa2) | -T(xb1,xc,xb2) | -E(xc,xa1,xc,xb1) | -E(xc,xa2,xc,xb2) |
-M(xa1,xm1,xb1) | -M(xa2,xm2,xb2) | T(xm1,xc,xm2).
%following defines a Krippenfigur (definition 7.23)
-T(xa1,xc,xa2) | -T(xb1,xc,xb2) | -E(xc,xa1,xc,xb1) | -E(xc,xa2,xc,xb2) |
-M(xa1,xm1,xb1) | -M(xa2,xm2,xb2) | KF(xa1,xm1,xb1,xc,xb2,xm2,xa2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | T(xa1,xc,xa2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) |  T(xb1,xc,xb2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | E(xc,xa1,xc,xb1).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | E(xc,xa2,xc,xb2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | M(xa1,xm1,xb1).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) |  M(xa2,xm2,xb2).
% following is Lemma 7.25
-E(xc,xa,xc,xb) | M(xa,midpoint(xa,xb),xb).
% following is Definition 8.1
-R(xa,xb,xc) | E(xa,xc,xa,s(xb,xc)).
R(xa,xb,xc) | -E(xa,xc,xa,s(xb,xc)).
% following is Satz 8.2
-R(xa,xb,xc) | R(xc,xb,xa).
% following is Satz 8.3
-R(xa,xb,xc) | xa = xb | -Col(xb,xa,xa1) | R(xa1,xb,xc).
% Following is Satz 8.4
-R(xa,xb,xc) | R(xa,xb,s(xb,xc)).
% following is Satz 8.5
R(x,y,y).
% following is Satz 8.6
-R(xa,xb,xc) | -R(xa1,xb,xc) | -T(xa,xc,xa1) | xb = xc.
% following is Satz 8.7
-R(xa,xb,xc) | -R(xa,xc,xb) | xb = xc.
% Following is Satz 8.8
-R(xa,xb,xa) | xa = xb.
% following is Satz 8.9
-R(xa,xb,xc) | -Col(xa,xb,xc) | xa=xb | xc=xb.
% following is Satz 8.10
-R(xa,xb,xc) | -E3(xa,xb,xc,xa1,xb1,xc1) | R(xa1,xb1,xc1).
% following is Definition 8.11 (i)
-perpAt(y,z,x,y1,z1) | Col(y,z,x).
-perpAt(y,z,x,y1,z1) | Col(y1,z1,x).
-perpAt(y,z,x,y1,z1) |y != z.
-perpAt(y,z,x,y1,z1) |y1 != z1.
-perpAt(y,z,x,y1,z1) | -Col(y,z,u) | -Col(y1,z1,v) | R(u,x,v).
perpAt(y,z,x,y1,z1) | y = z | y1 = z1 | -Col(y,z,x) | -Col(y1,z1,x) |
Col(y,z,f811(y,z,y1,z1,x)).
perpAt(y,z,x,y1,z1) | y = z | y1 = z1 | -Col(y,z,x) | -Col(y1,z1,x) |
Col(y1,z1,g811(y,z,y1,z1,x)).
perpAt(y,z,x,y1,z1) | y = z | y1 = z1 | -Col(y,z,x) | -Col(y1,z1,x) |
R(f811(y,z,y1,z1,x),x,g811(y,z,y1,z1,x)).
% following is Definition 8.11 (ii)
-perp(xp,xq,xp1,xq1) | perpAt(xp,xq,il(xp,xq,xp1,xq1),xp1,xq1).
perp(xp,xq,xp1,xq1) | -perpAt(xp,xq,x,xp1,xq1).
% Definition 8.11 (iii) disappears when we reduce to first-order
expressions.
% following is Satz 8.12a
-perpAt(x,y,z,u,v) | perpAt(u,v,z,x,y).
% following is Satz 8.12b
-perp(x,y,u,v) | perp(u,v,x,y).
% following is Satz 8.13a
-perpAt(xa,xb,x,xp,xq) | xa != xb.
-perpAt(xa,xb,x,xp,xq) | xp != xq.
-perpAt(xa,xb,x,xp,xq) | Col(xa,xb,x).
-perpAt(xa,xb,x,xp,xq) | Col(xp,xq,x).
-perpAt(xa,xb,x,xp,xq) | R(xa,x,xp).
-perpAt(xa,xb,x,xp,xq) | R(xb,x,xp).
-perpAt(xa,xb,x,xp,xq) | R(xa,x,xq).
-perpAt(xa,xb,x,xp,xq) | R(xb,x,xq).
% following is Satz 8.13b
-Col(xa,xb,x) | -Col(xp,xq,x) | xa = xb | xp = xq | -Col(xa,xb,u) |
-Col(xp,xq,v) | u = x |  v = x | -R(u,x,v) | perpAt(xa,xb,x,xp,xq).
% following is Satz 8.14a
-perp(xa,xb,xp,xq) | -Col(xa,xb,xp) | -Col(xa,xb,xq).
% following is Satz 8.14b
-perp(xa,xb,xp,xq) | -Col(xa,xb,xc) | -Col(xp,xq,xc) | perpAt(xa,xb,xc,xp,xq).
% following is Satz 8.14c
-perpAt(xa,xb,xc,xp,xq) | xc = il(xa,xb,xp,xq).
%following is Satz 8.15a
xa=xb | -Col(xa,xb,x) | -perp(xa,xb,xc,x) | perpAt(xa,xb,x,xc,x).
% following is Satz 8.15b
xa=xb | -Col(xa,xb,x) | perp(xa,xb,xc,x) | -perpAt(xa,xb,x,xc,x).
% following two are Satz 8.16a
xa = xb | -Col(xa,xb,x) | -Col(xa,xb,xu) | xu = x 
| -perp(xa,xb,xc,x) | -Col(xa,xb,xc).
xa = xb | -Col(xa,xb,x) | -Col(xa,xb,xc) | xu = x 
| -perp(xa,xb,xc,x) |  R(xc,x,xu).
% following is Satz8.16b
xa = xb | -Col(xa,xb,xp) | -Col(xa,xb,xq) | xq = xp 
| perp(xa,xb,xc,xp) | Col(xa,xb,xc) | -R(xc,xp,xq).
% following is Satz 8.18a
Col(xa,xb,xc) | -Col(xa,xb,xp) | -Col(xa,xb,xq) | -perp(xa,xb,xc,xp) | -perp(xa,xb,xc,xq) | xp = xq.
% following is Satz 8.18b
Col(xa,xb,xc) | Col(xa,xb,foot(xa,xb,xc)).
Col(xa,xb,xc) | perp(xa,xb,xc,foot(xa,xb,xc)).xs
% following is Lemma 8.20a
-R(xa,xb,xc) | - M(s(xa,xc),xp,s(xb,xc)) | R(xb,xa,xp).
% following is Lemma 8.20b
-R(xa,xb,xc) | - M(s(xa,xc),xp,s(xb,xc)) | -R(xb,xa,xp) | xb = xc | xa != xp. 


%Following is not among Szmielew's theorems because she used set theory for lines.  
% We prove it in perp1.in 
-perp(x,y,u,v) | perp(y,x,u,v).  

% following are some things we need, that are not in SST due to extensional treatment of lines.
% following is proved in ExtPerp.in (in Chapter 8) 
-Col(xa,xb,y) | -Col(xa,xb,z) | y=z | xa = xb 
| -perpAt(y,z,x,xc,xd) | perpAt(xa,xb,x,xc,xd).
% following is proved in ExtPerp2.in
-Col(xa,xb,y) | -Col(xa,xb,z) | y=z | xa = xb 
| -perp(y,z,xc,xd) | perp(xa,xb,xc,xd).
-Col(xa,xb,y) | -Col(xa,xb,z) | y=z | xa = xb 
| perp(y,z,xc,xd) | -perp(xa,xb,xc,xd).
% following is proved in ExtPerp3.in
xa = xb | xa = xc | xb = xc | xd = xc | xa = xd | -perp(xb,xa,xa,xc) 
| -Col(xa,xc,xd) | perp(xb,xa,xa,xd).
% following is proved in ExtPerp4.in
-perp(xa,xb,u,v) | perp(xa,xb,v,u).
% following is proved in ExtCol.in:
xa = xb | xp=xq | -Col(xa,xb,xp) | -Col(xa,xb,xq) 
| -Col(xa,xb,xt) | Col(xp,xq,xt). 
% following is proved in ExtCol2.in
xa = xb | xc = xd | -Col(xa,xb,xc) | -Col(xa,xb,xd) 
| -Col(xc,xd,xp) | Col(xa,xb,xp). 

% following proved in SideReflect.in  
-sameside(x,y,z) | sameside(s(u,x),s(u,y),s(u,z)).
% following is Satz 8.21
xa = xb |  perp(xa,xb,erect(xa,xb,xc),xa).
xa = xb |  Col(xa,xb,erectAux(xa,xb,xc)).
xa = xb | T(xc,erectAux(xa,xb,xc),erect(xa,xb,xc)).

% following is Satz 8.22
M(xa,midpoint(xa,xb),xb).

% following is Definition 9.1
xp = xq | Col(xp,xq,xa) | Col(xp,xq,xb) | -T(xa,xt,xb) | -Col(xp,xq,xt)
| opposite(xa,xp,xq,xb).
-opposite(xa,xp,xq,xb) | -Col(xp,xq,xa).
-opposite(xa,xp,xq,xb) | -Col(xp,xq,xb).
-opposite(xa,xp,xq,xb) | T(xa,il(xa,xb,xp,xq),xb).
-opposite(xa,xp,xq,xb) | Col(xp,xq,il(xa,xb,xp,xq)).
% following is Satz 9.2
-opposite(xa,xp,xq,xb) | opposite(xb,xp,xq,xa).
% following is Satz 9.5
-opposite(xa,xp,xq,xc) | -Col(xp,xq,xr) | -sameside(xa,xr,xb)
| opposite(xb,xp,xq,xc).
% following is Satz 9.6, outer Pasch
-T(xa,xc,xp) | -T(xb,xq,xc) | T(xa,op(xq,xb,xp,xc,xa),xb).
-T(xa,xc,xp) | -T(xb,xq,xc) | T(xp,xq,op(xq,xb,xp,xc,xa)).
% following is Defn 9.7, a and b are on the same side of Line(p,q).
-T(xa,xu,xc) | -T(xb,xv,xc) | -Col(xp,xq,xu) | -Col(xp,xq,xv) |xp=xq
| xc = xu | xc = xv | xa = xu | xb = xv | 
Col(xp,xq,xa) | Col(xp,xq,xb)  | samesideline(xa,xb,xp,xq). 
-samesideline(xa,xb,xp,xq) | Col(xp,xq,ss1(xa,xb,xp,xq)).   
-samesideline(xa,xb,xp,xq) | Col(xp,xq,ss2(xa,xb,xp,xq)).
-samesideline(xa,xb,xp,xq) | T(xa,ss1(xa,xb,xp,xq),ss3(xa,xb,xp,xq)).
-samesideline(xa,xb,xp,xq) | T(xb,ss2(xa,xb,xp,xq),ss3(xa,xb,xp,xq)).
-samesideline(xa,xb,xp,xq) | xa != ss1(xa,xb,xp,xq).
-samesideline(xa,xb,xp,xq) | xb != ss2(xa,xb,xp,xq).
-samesideline(xa,xb,xp,xq) | ss3(xa,xb,xp,xq) != ss1(xa,xb,xp,xq).
-samesideline(xa,xb,xp,xq) | ss3(xa,xb,xp,xq) != ss2(xa,xb,xp,xq). 
-samesideline(xa,xb,xp,xq) | xp != xq.
-samesideline(xa,xb,xp,xq) | -Col(xp,xq,xa).
-samesideline(xa,xb,xp,xq) | -Col(xp,xq,xb).
% Following is Satz 9.8
-opposite(xa,xp,xq,xc) | -opposite(xb,xp,xq,xc) | samesideline(xa,xb,xp,xq).
-opposite(xa,xp,xq,xc) | opposite(xb,xp,xq,xc) | -samesideline(xa,xb,xp,xq).

% A variant of Satz 9.8, Lemma 9.13e
-opposite(xc,xp,xq,xa) |  opposite(xc,xp,xq,xb) | -samesideline(xa,xb,xp,xq).  % plane separation theorem.
% You would think that this could replace 9.13e but it won't work for me:
% -opposite(xa,xp,xq,xb) | opposite(xb,xp,xq,xa).
% Following is Satz 9.9
-opposite(xa,xp,xq,xb) | -samesideline(xa,xb,xp,xq).


% Satz 9.13
-samesideline(u,v,x,y) | -samesideline(v,w,x,y) | samesideline(u,w,x,y).   

% Lemmas about samesideline that Szmielew doesn't need because
% she treats lines as sets of points.
%samesideline(x,y,u,v) | -samesideline(x,y,v,u).    % Lemma 9.13a
% following is Lemma 9.13b
%-T(xe,xd,xd1) | -samesideline(u,v,xd1,xe) | xd=xe | samesideline(u,v,xd,xe).
% following is Lemma 9.13c
-opposite(x,u,v,y) | opposite(y,u,v,x).
% following is Lemma 9.13d
-samesideline(x,y,u,v) | samesideline(y,x,u,v).


 

% Lemma 9.13f (a form of the plane separation theorem)
% This lemma is proved using the dimension axiom for n=2
xp = xq | Col(xp,xq,xs) | Col(xp,xq,xr) | opposite(xr,xp,xq,xs)
 | samesideline(xr,xs,xp,xq).
 


% Following is Satz 10.2a
xa = xb | Col(xa,xb,midpoint(u,reflect(xa,xb,u))).
xa = xb | u = reflect(xa,xb,u) | perp(xa,xb,u,reflect(xa,xb,u)).


% following is Satz 10.2b
xa = xb | -Col(xa,xb,midpoint(xp,xp1)) | -perp(xa,xb,xp,xp1)
| xp1 = reflect(xa,xb,xp).
xa = xb | -Col(xa,xb,midpoint(xp,xp1)) | xp != xp1 | xp1 = reflect(xa,xb,xp).


%xa = xb | Col(xa,xb,xp) | xp != reflect(xa,xb,xp).

% So the following is Satz 10.8d
xa = xb | -Col(xa,xb,xp) | xp = reflect(xa,xb,xp).

%xa = xb | -Col(xa,xb,z) | -perp(xa,xb,xp,z) | -E(xp,z,z,xp1) | xp1 =
% reflect(xa,xb,xp).
%xa = xb | -Col(xa,xb,xp) | xp = reflect(xa,xb,xp).
%xa = xb | Col(xa,xb,u) | perp(xa,xb,u,reflect(xa,xb,u)).

% following is the rest of definition 10.3
xa != xb | reflect(xa,xb,z) = s(xa,z).

% following is Satz 10.4
xa = xb |  reflect(xa,xb,xp) != xp1  | reflect(xa,xb,xp1) = xp.

% following is Satz 10.5
xa = xb | reflect(xa,xb,reflect(xa,xb,xp)) = xp.

% following is Satz 10.6
xa = xb | reflect(xa,xb,xp) != xp1 | xp = reflect(xa,xb,xp1).

% following is Satz 10.7
xa = xb | reflect(xa,xb,xp) != reflect(xa,xb,xq) | xp = xq.

% following is Satz 10.8a
xa = xb | reflect(xa,xb,xp) !=  xp | Col(xa,xb,xp).

% following is Satz 10.10
xa = xb | E(xp,xq,reflect(xa,xb,xp),reflect(xa,xb,xq)).
 



% Satz 10.12

-R(xa,xb,xc) | -R(xa1,xb1,xc1) | -E(xa,xb,xa1,xb1)
| -E(xb,xc,xb1,xc1) | E(xa,xc,xa1,xc1).

% Satz 10.14
reflect(u,v,xp) != xp1 | Col(u,v,xp) | opposite(xp,u,v,xp1).

% Satz 10.15
u = v | -Col(u,v,xa) | Col(u,v,xq) | perp(u,v, erectsameside(u,v,xa,xq),xa).
u = v | -Col(u,v,xa) | Col(u,v,xq)
| samesideline(erectsameside(u,v,xa,xq),xq,u,v).

% Satz 10.16a   (triangle existence)
xa=xb | Col(xa,xb,xc) | Col(xa1,xb1,xp) | -E(xa,xb,xa1,xb1)
| E3(xa,xb,xc,xa1,xb1,triangle(xa,xb,xc,xa1,xb1,xp)).
xa=xb | Col(xa,xb,xc) | Col(xa1,xb1,xp) | -E(xa,xb,xa1,xb1)
| samesideline( triangle(xa,xb,xc,xa1,xb1,xp),xp,xa1,xb1).

% Satz 10.16b   (triangle uniqueness)
Col(xa,xb,xc) | Col(xa1,xb1,xp) | -E(xa,xb,xa1,xb1)
| -E3(xa,xb,xc,xa1,xba1,xc1) | -E3(xa,xb,xc,xa1,xb1,xc2)
| -samesideline(xc1,xp,xa1,xb1) | -samesideline(xc2,xp,xa1,xb1) | xc1=xc2.

% Definition 11.2
congruent(xa,xb,xc,xd,xe,xf) | -T(xb,xa,xa1) | -E(xa,xa1,xe,xd) | -T(xb,xc,xc1)
| -E(xc,xc1,xe,xf) | -T(xe,xd,xd1) | -E(xd,xd1,xb,xa) | -T(xe,xf,xf1)
| -E(xf,xf1,xb,xc)    | -E(xa1,xc1,xd1,xf1) | xa=xb | xc=xb | xd=xe | xf=xe.

-congruent(xa,xb,xc,xd,xe,xf) |
E(ext(xb,xc,xe,xf),ext(xb,xa,xe,xd),ext(xe,xf,xb,xc),ext(xe,xd,xb,xa)).  
-congruent(xa,xb,xc,xd,xe,xf) | T(xb,xa,ext(xb,xa,xe,xd)).
-congruent(xa,xb,xc,xd,xe,xf) | T(xb,xc,ext(xb,xc,xe,xf)). 
-congruent(xa,xb,xc,xd,xe,xf) | T(xe,xd,ext(xe,xd,xb,xa)).
-congruent(xa,xb,xc,xd,xe,xf) | T(xe,xf,ext(xe,xf,xb,xc)).
-congruent(xa,xb,xc,xd,xe,xf) | xa != xb.
-congruent(xa,xb,xc,xd,xe,xf) | xc != xb.
-congruent(xa,xb,xc,xd,xe,xf) | xd != xe.
-congruent(xa,xb,xc,xd,xe,xf) | xf != xe. 

% following is Satz 11.3a (left-to-right direction of Satz 11.3)
-congruent(xa,xb,xc,xd,xe,xf) |  sameside(ext(xb,xa,xe,xd),xb,xa).  
-congruent(xa,xb,xc,xd,xe,xf) |  sameside(ext(xb,xc,xe,xf),xb,xc).   
-congruent(xa,xb,xc,xd,xe,xf) |  sameside(ext(xe,xd,xb,xa),xe,xd).   
-congruent(xa,xb,xc,xd,xe,xf) |  sameside(ext(xe,xf,xb,xc),xe,xf).  
-congruent(xa,xb,xc,xd,xe,xf) |  
E3(ext(xb,xa,xe,xd),xb,ext(xb,xc,xe,xf),ext(xe,xd,xb,xa),xe,ext(xe,xf,xb,xc)).  
 
% following is Satz 11.4, i.e. (3)=> (4) from page 95
-sameside(xa1,xb,xa2) |
-sameside(xc1,xb,xc2) | 
-sameside(xd1,xe,xd2) |
-sameside(xf1,xe,xf2) | 
-E3(xa1,xb,xc1,xd1,xe,xf1) | 
xa=xb | xc=xb | xd=xe | xf=xe |
-sameside(xa2,xb,xa) |
-sameside(xc2,xb,xc) |
-sameside(xd2,xe,xd) |
-sameside(xf2,xe,xf) |  
-E(xb,xa2,xe,xd2) | 
-E(xb,xc2,xe,xf2) |
E(xa2,xc2,xd2,xf2).      

% following is Satz 11.3d, right-to-left direction of Satz 11.3
-sameside(xa1,xb,xa) | -sameside(xc1,xb,xc) 
| -sameside(xd1,xe,xd)| -sameside(xf1,xe,xf) 
| -E3(xa1,xb,xc1,xd1,xe,xf1) | congruent(xa,xb,xc,xd,xe,xf).    

% following is Satz11.4B.in
-E(x,y,x1,y1) | -E(x,z,x1,z1) | -sameside(y,x,z) 
| -sameside(y1,x1,z1) |   E(y,z,y1,z1).

% Axiom A9 (upper dimension)
xp=xq | -E(xa,xp,xa,xq) | -E(xb,xp,xb,xq) | -E(xc,xp,xc,xq) 
| T(xa,xb,xc) | T(xb,xc,xa) | T(xc,xa,xb).
 
% Axiom A10:

-T(xa,xd,xt) | -T(xb,xd,xc) | xa=xd | T(xa,xb,e1(xa,xb,xc,xd,xt)).     
-T(xa,xd,xt) | -T(xb,xd,xc) | xa=xd | T(xa,xc,e2(xa,xb,xc,xd,xt)).  
-T(xa,xd,xt) | -T(xb,xd,xc) | xa=xd | T(e1(xa,xb,xc,xd,xt),xt,e2(xa,xb,xc,xd,xt)).
   


% definition 12.7  (echt parallel)  Note, we use 12.7 rather than 12.2,
% following the suggestion given in SST for those working with points only, 
% and thus avoiding the introduction of Cp.
 
 

-parallel(xA,yA,xB,yB) | xA != yA.  
-parallel(xA,yA,xB,yB) | xB != yB.  
-parallel(xA,yA,xB,yB) |  samesideline(xB,yB,xA,yA).  
 -parallel(xA,yA,xB,yB) | -Col(xA,yA,x) | -Col(xB,yB,x).  



xA=yA | xB=yB  | parallel(xA,yA,xB,yB) | 
   Col(xA,yA,e5(xA,yA,xB,yB)) | -samesideline(xB,yB,xA,yA).     
xA=yA | xB=yB  | parallel(xA,yA,xB,yB) |
    Col(xB,yB,e5(xA,yA,xB,yB)) | -samesideline(xB,yB,xA,yA).

% Satz 12.5


% Definition 12.3  (weakly parallel)  
-parallel(xA,yA,xB,yB) | weakparallel(xA,yA,xB,yB).     
-Col(xA,yA,xB) | -Col(xA,yA,yB) | weakparallel(xA,yA,xB,yB).
-weakparallel(xA,yA,xB,yB) | parallel(xA,yA,xB,yB) | Col(xA,yA,xB).
-weakparallel(xA,yA,xB,yB) | parallel(xA,yA,xB,yB) | Col(xA,yA,yB).  

% Satz 12.6
-Col(xB,yB,xb) | xB = yB |  -Col(xB,yB,xb1) | samesideline(xb,xb1,xA,yA) | -parallel(xA,yA,xB,yB).
 
 
 
 
end_of_list.  


list(hints2).  
Col(x,y,z)|opposite(z,x,y,reflect(x,y,z)).
s(x,x)=x.
sameside(t,q,t).
-E(p1,p2,x,x).
-opposite(a,p1,p2,x).
E(p2,c2,p1,a).
T(p1,p2,c2).
c!=c2|c2=a.
s(a,c)=c2.
M(c2,a,c).
Col(t,q,t).
Col(p1,p2,c2).
-E(p2,c2,p1,c2).
c!=c2.
c!=a.
T(c2,a,c).
-Col(r,cs,t).
s(a,c)!=a.
Col(c2,a,c).
c2!=a.
Col(p1,p2,c).
-Col(r,cs,c2).
-Col(r,cs,c).
samesideline(t,c2,r,cs).
opposite(c2,r,cs,c).
samesideline(c2,t,r,cs).
opposite(t,r,cs,c).
Col(r,cs,b2).
T(t,b2,c).
-T(c,b2,t).
T(c,b2,t).

end_of_list.



list(passive).   
%-samesideline(r,cs,t,q) | $ANS(1).
%-samesideline(p1,p2,t,q)| $ANS(2).
%-samesideline(c2,t,r,cs)| $ANS(3).
%-samesideline(t,c2,r,cs)| $ANS(4).
%-T(c,a,c2)| $ANS(8).
%-T(c2,a,c)| $ANS(9).
%-opposite(c,r,cs,c2)| $ANS(10).
%-opposite(t,r,cs,c)| $ANS(11).
%-T(t,b2,c) | $ANS(13).
%-Col(r,cs,b2) | $ANS(14).
%-Col(p1,p2,c) | $ANS(15).
%-Col(p1,p2,c2) | $ANS(16).
%-samesideline(r,a,t,q) | $ANS(20).
%-samesideline(a,c2,t,q) | $ANS(21).
%Col(r,cs,t) | $ANS(22).
%Col(r,cs,c) | $ANS(23).
end_of_list.
    
 

list(sos). 
x=x. 
t != q.    % Line(t,q) is line A
-Col(t,q,a).   % a not on line A
r != cs.     % Line(r,cs) is line B
Col(r,cs,a).   % a is on line B
p1 != p2.     % Line(p1,p2) is line C
Col(p1,p2,a).  % a is on line C
-Col(r,cs,p1) | -Col(r,cs,p2).  %C does not coincide with B 
parallel(t,q,r,cs).
parallel(t,q,p1,p2).  
-T(xc,xb,t) | -Col(p1,p2,xc) | -Col(r,cs,xb) | xc = a.

% The diagram:

c2 = ext(p1,p2,p1,a).   % Then c2 is on line C and not equal to a. 
c = s(a,c2). 
b2 = il(t,c,r,cs).
b = il(t,c2,r,cs). 


% It will do either case alone.
 -opposite(t,r,cs,c2).

     
 
 

 
 
end_of_list.

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