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NCABC
NCDEF
NOEAABCDEF
NOAODEFABC
EQBA  assumption
 EQAB  lemma:equalitysymmetric
 COABC defn:collinear
NEBA  reductio
EQBC assumption
 COABC  defn:collinear
NEBC reductio
COBAC assumption
 COABC  lemma:collinearorder
NCBAC reductio
ANRABAJ+EAGBJDEF+SSGCBA proposition:23C
RABAJ
EAGBJDEF
SSGCBA
NCBAG   defn:sameside
EQBG assumption
 COBAG defn:collinear
NEBG reductio
EQAG assumption
 COBAG defn:collinear
NEAG reductio
EADEFGBJ  lemma:equalanglessymmetric
RABJA  lemma:ray5
EQGG    cn:equalityreflexive
RABGG  lemma:ray4
EADEFGBA  lemma:equalangleshelper
NCGBA  lemma:equalanglesNC
COABG assumption
 COBAG lemma:collinearorder
NCABG reductio
EAGBADEF  lemma:equalanglessymmetric
EAABGGBA   lemma:ABCequalsCBA
EAABGDEF  lemma:equalanglestransitive

COABG  assumption
 COGBA  lemma:collinearorder
NCABG reductio
EQGA  assumption
 EQAG   lemma:equalitysymmetric
 COABG  defn:collinear
NEGA  reductio
ANBEGAP+EEAPGA lemma:extension
BEGAP
EQAA   cn:equalityreflexive
COBAA   defn:collinear
COBAG   assumption
 COGBA  lemma:collinearorder
NCBAG   reductio
SSCGBA   lemma:samesidesymmetric
OSGBAP   defn:oppositeside
OSCBAP   lemma:planeseparation
ANBECRP+COBAR+NCBAC  defn:oppositeside
BECRP
BEPRC   axiom:betweennesssymmetry
NCBAC
cases AOABCDEF:OSGBCA|NOOSGBCA
 case 1:OSGBCA
  ANBEGHA+COBCH+NCBCG defn:oppositeside
  BEGHA
  COBCH
  BEAHG  axiom:betweennesssymmetry
  RABAA  lemma:ray4
  RABGG
  COABH  assumption
   EQBH  assumption
    BEABG  cn:equalitysub
    COABG  defn:collinear
    COGBA  lemma:collinearorder
   NEBH  reductio
   NEHB  lemma:inequalitysymmetric
   COHBA  lemma:collinearorder
   COHBC  lemma:collinearorder
   COBAC   lemma:collinear4
   COABC   lemma:collinearorder
  NCABH  reductio
  EAABHABH  lemma:equalanglesreflexive
  AOABHABG  defn:anglelessthan
  EAGBAABG  lemma:ABCequalsCBA
  AOABHGBA  lemma:angleorderrespectscongruence
  COHBA   assumption
   COABH  lemma:collinearorder
  NCHBA   reductio
  EAHBAABH  lemma:ABCequalsCBA
  AOHBAGBA  lemma:angleorderrespectscongruence2
  AOHBADEF  lemma:angleorderrespectscongruence
  EAABHHBA  lemma:ABCequalsCBA
  AOABHDEF  lemma:angleorderrespectscongruence2
  RABAA   lemma:ray4
  SSGCBA  
  SSCGBA  lemma:samesidesymmetric
  BEAHG
  RAAGH   lemma:ray4
  EQAA     cn:equalityreflexive
  COBAA    defn:collinear
  SSCHBA   lemma:sameside2
  BECBH  assumption
   EQBB   cn:equalityreflexive
   COBAB  defn:collinear
   OSCBAH  defn:oppositeside
   OSHBAC  lemma:oppositesidesymmetric
   OSCBAC  lemma:planeseparation
   ANBECMC+COBAM+NCBAC defn:oppositeside
   BECMC
   NOBECMC  axiom:betweennessidentity
  NOBECBH  reductio
  COBCH
  OREQBC|EQBH|EQCH|BECBH|BEBCH|BEBHC  defn:collinear
  cases RABCH:EQBC|EQBH|EQCH|BECBH|BEBCH|BEBHC
   case 1:EQBC
    COABC defn:collinear
    NORABCH  assumption
     NCABC
    RABCH  reductio
   qedcase
   case 2:EQBH
    COBHA  defn:collinear
    NORABCH assumption
     COBHA assumption
      COHBA  lemma:collinearorder
      NCHBA
     NCBHA reductio
    RABCH reductio
   qedcase
   case 3:EQCH
    EQHH  cn:equalityreflexive
    cases RABCH:EQBH|NEBH
     case 1:EQBH
      COBHA  defn:collinear
	  NORABCH assumption
	   COBHA assumption
	    COHBA  lemma:collinearorder
	    NCHBA
	   NCBHA reductio
	  RABCH reductio
	 qedcase
	 case 2:NEBH
      RABHH  lemma:ray4
      RABCH  cn:equalitysub
     qedcase
    RABCH cases
   qedcase
   case 4:BECBH
    NORABCH assumption
     NOBECBH
    RABCH reductio
   qedcase
   case 5:BEBCH
    NEBC  lemma:betweennotequal
    RABCH  lemma:ray4
   qedcase
   case 6:BEBHC
    NEBC  lemma:betweennotequal
    RABCH lemma:ray4
   qedcase
  RABCH cases
  EAABCABC  lemma:equalanglesreflexive
  EAABCABH  lemma:equalangleshelper
  AOABCDEF  lemma:angleorderrespectscongruence2
 qedcase
 case 2: NOOSGBCA
  COBAR
  OREQBA|EQBR|EQAR|BEABR|BEBAR|BEBRA defn:collinear
  cases AOABCDEF:EQBA|EQBR|EQAR|BEABR|BEBAR|BEBRA
  case 1:EQBA
   NOAOABCDEF assumption
     NEBA
   AOABCDEF reductio
  qedcase
  case 2:EQBR
   BEGAP
   BECRP axiom:betweennesssymmetry
   COCPG assumption
    COCRP defn:collinear
    COCBP cn:equalitysub
    COGAP defn:collinear
    COGPA lemma:collinearorder
    COGPC lemma:collinearorder
    NEGP  lemma:betweennotequal
    COPCA lemma:collinear4
    COPCB lemma:collinearorder
    NECP lemma:betweennotequal
    NEPC lemma:inequalitysymmetric
    COCAB lemma:collinear4
    COABC lemma:collinearorder
    NCABC
   NCCPG reductio
   ANBECQA+BEGQR  postulate:Pasch-inner
   BECQA
   BEGQR 
   BEGQB  cn:equalitysub
   BEBQG  axiom:betweennesssymmetry
   NEBQ  lemma:betweennotequal
   NEBG   lemma:betweennotequal
   RABQG  lemma:ray4
   RABGQ  lemma:ray5
   EQQQ   cn:equalityreflexive
   EQAA   cn:equalityreflexive
   EQCC   cn:equalityreflexive
   RABAA  lemma:ray4
   RABCC  lemma:ray4
   RABGG  lemma:ray4
   RABQQ  lemma:ray4
   NCABG
   EEAQAQ  cn:congruencereflexive
   EEBQBQ  cn:congruencereflexive
   EEBABA  cn:congruencereflexive
   EAABGABQ  defn:equalangles
   BEAQC  axiom:betweennesssymmetry
   AOABGABC  defn:anglelessthan
   EAABGDEF
   EADEFABG  lemma:equalanglessymmetric
   AODEFABC  lemma:angleorderrespectscongruence2
   NOAOABCDEF assumption
     NOAODEFABC
   AOABCDEF reductio
  qedcase
  case 3:EQAR
   NOAOABCDEF assumption
    BEPAG   axiom:betweennesssymmetry
    BEPAC  cn:equalitysub
    EQGG  cn:equalityreflexive
    RABGG lemma:ray4
    EQAA  cn:equalityreflexive
    RABAA  lemma:ray4
    EQCC  cn:equalityreflexive
    RABCC  lemma:ray4
    EADEFABG  lemma:equalanglessymmetric
    BEAGC  assumption   
     EAABGABG lemma:equalanglesreflexive
     AOABGABC defn:anglelessthan  
     AODEFABC  lemma:angleorderrespectscongruence2
     NOAODEFABC
    NOBEAGC reductio
    BEACG  assumption
     EAABCABC lemma:equalanglesreflexive
     AOABCABG defn:anglelessthan
     AOABCDEF lemma:angleorderrespectscongruence
    NOBEACG  reductio
    EQCG  lemma:outerconnectivity
    EAABCABC lemma:equalanglesreflexive
    EAABGABC cn:equalitysub
    EAABCABG lemma:equalanglessymmetric
    EAABCDEF  lemma:equalanglestransitive
    EAABCDEF
   AOABCDEF reductio
  qedcase
  case 4:BEABR
   BERBA   axiom:betweennesssymmetry
   BECRP
   BEABR  axiom:betweennesssymmetry
   COCPA assumption
    COCRP  defn:collinear
    COCPR  lemma:collinearorder
    NECP  lemma:betweennotequal
    COPAR lemma:collinear4
    CORBA defn:collinear
    CORAB lemma:collinearorder
    CORAP lemma:collinearorder
    NERA  lemma:betweennotequal
    COABP lemma:collinear4
    COPAB lemma:collinearorder
    COGAP defn:collinear
    COPAG lemma:collinearorder
    NEAP lemma:betweennotequal
    NEPA lemma:inequalitysymmetric
    COABG lemma:collinear4
    NCABG
   NCCPA reductio
   ANBEAMP+BECBM  postulate:Pasch-outer
   BEAMP
   BECBM 
   BEGAP
   BEPAG   axiom:betweennesssymmetry
   BEPMA   axiom:betweennesssymmetry
   BEMAG lemma:3.6a
   BEGAM   axiom:betweennesssymmetry
   COCMG  assumption
    BEPMA  axiom:betweennesssymmetry
    BEPAG  axiom:betweennesssymmetry
    BEPMG  lemma:3.6b
    COPMG  defn:collinear
    COMGP  lemma:collinearorder
    COMGC lemma:collinearorder
    NEMG lemma:betweennotequal
    COGPC lemma:collinear4
    COPAG  defn:collinear
    COGPA lemma:collinearorder
    NEPG  lemma:betweennotequal
    NEGP lemma:inequalitysymmetric
    COPCA  lemma:collinear4
    COCPA  lemma:collinearorder
    NCCPA
   NCCMG reductio
   ANBECQA+BEGQB postulate:Pasch-inner
   BECQA
   BEGQB 
   BEBQG  axiom:betweennesssymmetry
   NEBQ   lemma:betweennotequal
   NEBG   lemma:betweennotequal
   RABQG  lemma:ray4
   RABGQ  lemma:ray5
   EQQQ   cn:equalityreflexive
   EQAA   cn:equalityreflexive
   EQCC   cn:equalityreflexive
   RABAA  lemma:ray4
   RABCC  lemma:ray4
   RABGG  lemma:ray4
   RABQQ  lemma:ray4
   NCABG
   EEAQAQ  cn:congruencereflexive
   EEBQBQ  cn:congruencereflexive
   EEBABA  cn:congruencereflexive
   EAABGABQ  defn:equalangles
   BEAQC  axiom:betweennesssymmetry
   AOABGABC  defn:anglelessthan
   EAABGDEF
   EADEFABG  lemma:equalanglessymmetric
   AODEFABC  lemma:angleorderrespectscongruence2
   NOAOABCDEF assumption
     NOAODEFABC
   AOABCDEF reductio
  qedcase
  case 5:BEBAR
   COPCB  assumption
    COBAR  defn:collinear
    COPRC  defn:collinear
    COPCR lemma:collinearorder
    NEPC  lemma:betweennotequal
    COCBR lemma:collinear4
    CORBC lemma:collinearorder
    CORBA lemma:collinearorder
    NEBR  lemma:betweennotequal
    NERB  lemma:inequalitysymmetric
    COBCA lemma:collinear4
    COABC lemma:collinearorder
    NCABC
   NCPCB  reductio
   ANBEBQC+BEPAQ  postulate:Pasch-outer
   BEBQC
   COBCQ   defn:collinear
   EQGQ   assumption
    BEBGC cn:equalitysub
    RABCG  lemma:ray4
    RABAA  lemma:ray4
    RABGG  lemma:ray4
    EEAGAG cn:congruencereflexive
    EEBGBG  cn:congruencereflexive
    EEBABA  cn:congruencereflexive
    NCABG
    EAABGABC  defn:equalangles
    EAABCABG  lemma:equalanglessymmetric
    EAABGDEF
    EAABCDEF  lemma:equalanglestransitive
    NOEAABCDEF
   NEGQ   reductio
   COBCG  assumption
    BEPAQ
    BEPAG  axiom:betweennesssymmetry
    RAAGQ  defn:ray
    COAGQ  lemma:rayimpliescollinear
    COCBG  lemma:collinearorder
    COCBQ  lemma:collinearorder
    NEBC   lemma:betweennotequal
    NECB   lemma:inequalitysymmetric
    EQBB   cn:equalityreflexive
    COCBB  defn:collinear
    COGQB  lemma:collinear5
    COQGB  lemma:collinearorder
    COQGA  lemma:collinearorder   
    NEQG   lemma:inequalitysymmetric 
    COGBA  lemma:collinear4
    COABG  lemma:collinearorder
   NCBCG  reductio
   BEAQG  assumption
    BEGQA   axiom:betweennesssymmetry
    OSGBCA  defn:oppositeside
   NOBEAQG  reductio
   RABCQ   lemma:ray4
   RABAA    lemma:ray4
   BEAGQ   assumption
    EAABGABG  lemma:equalanglesreflexive
    AOABGABC  defn:anglelessthan
    EADEFABG  lemma:equalanglessymmetric
    AODEFABC  lemma:angleorderrespectscongruence2
   NOBEAGQ  reductio
   BEPAQ
   BEPAG  axiom:betweennesssymmetry
   NOBEAGQ
   NOBEAQG
   EQGQ   lemma:outerconnectivity
   NOAOABCDEF assumption
    NEGQ  
   AOABCDEF reductio 
  qedcase
  case 6:BEBRA 
   NOAOABCDEF assumption
    BEPAG   axiom:betweennesssymmetry
    BEBRA  
    COPGB  assumption
     COPAG  defn:collinear
     COPGA  lemma:collinearorder
     NEPG  lemma:betweennotequal
     COGBA  lemma:collinear4
     COABG  lemma:collinearorder
     NCABG
    NCPGB reductio
    ANBEBQG+BEPRQ  postulate:Pasch-outer
    BEBQG
    NEBQ  lemma:betweennotequal
    RABQG   lemma:ray4
    BERCQ  assumption
     RABAR  lemma:ray4
     RABGQ  lemma:ray4
     EAABCABC lemma:equalanglesreflexive
     AOABCABG  defn:anglelessthan
     EAABGDEF
     EADEFABG lemma:equalanglessymmetric
     AOABCDEF  lemma:angleorderrespectscongruence
    NOBERCQ  reductio
    BERQC   assumption
     EQAA   cn:equalityreflexive
     RABAA  lemma:ray4
     RABQG   lemma:ray4
     EQGG   cn:equalityreflexive
     RABGG  lemma:ray4
     EEBABA  cn:congruencereflexive
     EEBGBG  cn:congruencereflexive
     EEAGAG  cn:congruencereflexive 
     NCABG
     EAABGABQ defn:equalangles
     RABAR  lemma:ray4
     EQCC    cn:equalityreflexive
     RABCC   lemma:ray4
     AOABGABC  defn:anglelessthan
     EADEFABG  lemma:equalanglessymmetric
     AODEFABC  lemma:angleorderrespectscongruence2
     NOAODEFABC 
    NOBERQC   reductio
    BEPRQ
    BEPRC
    EQQC    lemma:outerconnectivity
    EQCC   cn:equalityreflexive
    RABCC  lemma:ray4
    RABCG   cn:equalitysub
    EQAA   cn:equalityreflexive
    RABAA  lemma:ray4
    EQGG   cn:equalityreflexive
    RABGG  lemma:ray4
    EEBABA  cn:congruencereflexive
    EEBGBG  cn:congruencereflexive
    EEAGAG  cn:congruencereflexive 
    EAABCABG  defn:equalangles
    EAABCDEF  lemma:equalanglestransitive 
    EAABCDEF
    AOABCDEF reductio
  qedcase
 AOABCDEF cases
 qedcase 
AOABCDEF cases