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BEAEB
BECFD
EAAEFEFD
OSAEFD
NEAB   lemma:betweennotequal
NECD   lemma:betweennotequal
ANBEAHD+COEFH+NCEFA  defn:oppositeside
BEAHD
COEFH
NCEFA 
COAEB  defn:collinear
NEAE  lemma:betweennotequal
NEEB  lemma:betweennotequal
COCFD  defn:collinear
NECF  lemma:betweennotequal
NEFD  lemma:betweennotequal
EAEFDAEF  lemma:equalanglessymmetric
NCEFD   defn:equalangles
NEEF  lemma:angledistinct
NEFE  lemma:inequalitysymmetric
MEABCD  assumption
 ANNEAB+NECD+COABG+COCDG  defn:meet
 COABG
 COCDG
 COBAG  lemma:collinearorder
 COBAE  lemma:collinearorder
 NEBA   lemma:inequalitysymmetric
 COAGE  lemma:collinear4
 COAEG  lemma:collinearorder
 NCAEF  defn:equalangles
 EQAE  assumption
  COAEF  defn:collinear
 NEAE  reductio
 EQFF   cn:equalityreflexive
 RAEFF   lemma:ray4
 SUAEFFB  defn:supplement
 EAEFDAEF  lemma:equalanglessymmetric
 EQEE  cn:equalityreflexive
 RAFEE  lemma:ray4
 BEDFC  axiom:betweennesssymmetry
 SUDFEEC  defn:supplement
 EAEFDDFE  lemma:ABCequalsCBA
 EAAEFDFE  lemma:equalanglestransitive
 EAFEBEFC  lemma:supplements
 EABEFCFE  lemma:equalanglesflip
 BEAEG assumption   
% The case Euclid does prove
  EQEE     cn:equalityreflexive
  COEFE    defn:collinear
  NCEFA
  BEGEA    axiom:betweennesssymmetry 
  NCEFD
  NEFD
  COCDG
  COCFD  defn:collinear
  COCDF  lemma:collinearorder
  NECD   lemma:betweennotequal
  CODGF  lemma:collinear4
  COGFD  lemma:collinearorder
  EQFG   assumption
   COAEF   cn:equalitysub
   COEFA   lemma:collinearorder
  NEFG   reductio
  NEGF   lemma:inequalitysymmetric
  COEFG  assumption
   COGFE  lemma:collinearorder
   COFED  lemma:collinear4
   COEFD   lemma:collinearorder
   NCEFD
  NCEFG  reductio
  BEDHA    axiom:betweennesssymmetry
  COEFH
  SSDGEF   defn:sameside
  SSGDEF   lemma:samesidesymmetric
  EQFF   cn:equalityreflexive
  COEFF   defn:collinear
  NCEFD
  BEDFC   axiom:betweennesssymmetry
  OSDEFC   defn:oppositeside
  OSGEFC   lemma:planeseparation
  ANBEGRC+COEFR+NCEFG  defn:oppositeside
  BEGRC
  COEFR
  NCEFG
  NEFR assumption 
   COEFR
   EQFF   cn:equalityreflexive
   COEFF  defn:collinear
   COEFR
   COCDF
   COGRC  defn:collinear
   COCDG
   COCGD  lemma:collinearorder
   COCGR  lemma:collinearorder
   NEGC   lemma:betweennotequal
   NECG   lemma:inequalitysymmetric
   COGDR  lemma:collinear4
   COGCR  lemma:collinearorder
   COGCD  lemma:collinearorder
   NEGC  lemma:inequalitysymmetric
   COCRD  lemma:collinear4
   COCDR  lemma:collinearorder
   COCDF   defn:collinear
   COEFR
   EQEE   cn:equalityreflexive
   COEFE  defn:collinear
   NERF  lemma:inequalitysymmetric
   COCGR lemma:collinearorder
   COCDF  lemma:collinearorder
   COCDG
   CODFG lemma:collinear4
   CODFC lemma:collinearorder
   NEFD  lemma:betweennotequal
   NEDF  lemma:inequalitysymmetric
   COFGC lemma:collinear4
   COCGF lemma:collinearorder
   COCGD lemma:collinearorder
   CORFD lemma:collinear5
   CORFE  lemma:collinearorder
   COFDE lemma:collinear4
   COEFD  lemma:collinearorder
   NCEFD
  EQFR reductio
  BEGFC cn:equalitysub
  COEGF  assumption
   COEFG  lemma:collinearorder
   NCEFG
  NCEGF reductio
  TREGF   defn:triangle  
  AOGEFEFC  proposition:16
  EAFEBEFC   
  AOGEFFEB  lemma:angleorderrespectscongruence
  EQFF  cn:equalityreflexive
  RAEFF   lemma:ray4
  RAEGB   defn:ray
  COGEF  assumption
   COEGF lemma:collinearorder
  NCGEF  reductio
  EAGEFGEF  lemma:equalanglesreflexive
  EAGEFBEF  lemma:equalangleshelper
  NCBEF   defn:equalangles
  EABEFFEB  lemma:ABCequalsCBA
  EAGEFFEB  lemma:equalanglestransitive
  EAFEBGEF  lemma:equalanglessymmetric
  AOFEBFEB  lemma:angleorderrespectscongruence2
  NOAOFEBFEB  lemma:angletrichotomy
 NOBEAEG reductio
 RAEAG   assumption
  EQFF     cn:equalityreflexive
  RAEFF   lemma:ray4
  RAEGA    lemma:ray5
  EAEFDAEF  lemma:equalanglessymmetric
  EAEFDGEF  lemma:equalangleshelper
  EAGEFEFD  lemma:equalanglessymmetric
  BEBEA    axiom:betweennesssymmetry
  ORBEEAG|EQGA|BEEGA  lemma:ray1
  cases BEBEG:BEEAG|EQGA|BEEGA
   case 1:BEEAG
    BEBEA
    BEBEG  lemma:3.7b
   qedcase
   case 2:EQGA
    BEBEG  cn:equalitysub
   qedcase
   case 3:BEEGA
    BEBEG axiom:innertransitivity
   qedcase
  BEBEG cases  
  BEGEB axiom:betweennesssymmetry
  EQEE     cn:equalityreflexive
  COEFE    defn:collinear 
  COEFG  assumption
   COABG
   COBAG  lemma:collinearorder
   COAEB  defn:collinear
   COBAE  lemma:collinearorder
   COAGE  lemma:collinear4
   COGEA  lemma:collinearorder
   COGEF  lemma:collinearorder
   NEEG   lemma:betweennotequal
   NEGE   lemma:inequalitysymmetric
   COEAF  lemma:collinear4
   COEFA  lemma:collinearorder
   NCEFA
  NCEFG  reductio
  NCEFA
  SSAGEF    defn:sameside
  SSGAEF    lemma:samesidesymmetric
  OSAEFD
  OSGEFD    lemma:planeseparation
  ANBEGPD+COEFP+NCEFG defn:oppositeside
  BEGPD
  COGPD  defn:collinear
  COEFP
  NEPF  assumption
   NEGD  lemma:betweennotequal
   COGDP lemma:collinearorder
   COCDG
   COCFD  defn:collinear
   COCDF  lemma:collinearorder
   CODGF  lemma:collinear4
   COGDF  lemma:collinearorder
   CODPF  lemma:collinear4
   COPFD  lemma:collinearorder
   COPFE  lemma:collinearorder
   COFDE  lemma:collinear4
   COFDE  assumption
    COEFD  lemma:collinearorder
   NCFDE  reductio
  EQPF    reductio
  BEGFD   cn:equalitysub
  RAEAG 
  COFEA  assumption
   COEFA  lemma:collinearorder
  NCFEA  reductio
  EAFEAFEA  lemma:equalanglesreflexive
  EAFEAFEG  lemma:equalangleshelper
  EAFEGFEA  lemma:equalanglessymmetric
  NCFEG   defn:equalangles
  BEGFD
  COEGF  assumption
   COFEG lemma:collinearorder
  NCEGF  reductio
  TREGF  defn:triangle
  AOGEFEFD  proposition:16
  AOEFDEFD  lemma:angleorderrespectscongruence2
  NOAOEFDEFD lemma:angletrichotomy
 NORAEAG reductio
 OREQAE|EQAG|EQEG|BEEAG|BEAEG|BEAGE   defn:collinear
 cases NOMEABCD:EQAE|EQAG|EQEG|BEEAG|BEAEG|BEAGE 
  case 1:EQAE
   MEABCD assumption
    NEAE  
   NOMEABCD reductio
  qedcase
  case 2:EQAG
   COAFE  assumption
    COEFA  lemma:collinearorder
   NCAFE  reductio
   TRAFE  defn:triangle
   NEHF   assumption
    COEFH
    COCDG
    COCDA  cn:equalitysub
    COCDF  lemma:collinearorder
    NECD
    CODGF  lemma:collinear4
    CODAF  cn:equalitysub
    COAHD  defn:collinear
    CODAH  lemma:collinearorder
    NEAD  lemma:betweennotequal
    NEDA   lemma:inequalitysymmetric
    COAFH  lemma:collinear4
    COHFA  lemma:collinearorder
    COEFH
    COHFE  lemma:collinearorder
    COFAE  lemma:collinear4
    COEFA  lemma:collinearorder
    NCEFA
   EQHF   reductio
   BEAFD   cn:equalitysub
   COEAF  assumption
    COEFA   lemma:collinearorder
   NCEAF  reductio
   TREAF    defn:triangle
   AOAEFEFD  proposition:16
   EAEFDAEF  lemma:equalanglessymmetric 
   AOEFDEFD  lemma:angleorderrespectscongruence2
   MEABCD assumption
    NOAOEFDEFD  lemma:angletrichotomy
   NOMEABCD reductio
  qedcase
  case 3:EQEG
   COCDE  cn:equalitysub
   COCDF  lemma:collinearorder
   CODEF lemma:collinear4
   COEFD  lemma:collinearorder
   COEFH
   NEEF  assumption
    COFDH lemma:collinear4
    CODHF lemma:collinearorder
    COAHD  defn:collinear
    CODHA  lemma:collinearorder
    NEHD  lemma:betweennotequal
    NEDH  lemma:inequalitysymmetric
    COHFA lemma:collinear4
    COEFH 
    COHFE  lemma:collinearorder
    NEHF  assumption
     COFAE lemma:collinear4
     COEFA lemma:collinearorder
     NCEFA
    EQHF  reductio
    COAHD  defn:collinear
    COAFD  cn:equalitysub
    CODFA lemma:collinearorder
    CODFC lemma:collinearorder
    NEHD  lemma:betweennotequal
    NEDH  lemma:inequalitysymmetric
    NEDF  cn:equalitysub
    COFAC lemma:collinear4
    COCFA lemma:collinearorder
    COCDG
    CODCG  lemma:collinearorder
    COCDF  lemma:collinearorder
    CODCF  lemma:collinearorder
    NEDC  lemma:inequalitysymmetric
    COCGF  lemma:collinear4
    COCFG  lemma:collinearorder
    NECF  assumption
     COFAG lemma:collinear4
     COFAE cn:equalitysub
     COEFA lemma:collinearorder
    EQCF  reductio
    COCDE cn:equalitysub
    EQCH  cn:equalitytransitive
    COAHD  defn:collinear
    COACD  cn:equalitysub
    COCDA  lemma:collinearorder
    COFDA  cn:equalitysub
    COCDE  cn:equalitysub
    COFDE  cn:equalitysub
    CODFE  lemma:collinearorder
    CODFA  lemma:collinearorder
    NEDF   cn:equalitysub
    COFEA  lemma:collinear4
    COEFA  lemma:collinearorder
   EQEF  reductio
   COEFA  defn:collinear
   MEABCD assumption
    NCEFA
   NOMEABCD reductio
  qedcase
  case 4:BEEAG
   NEEA  lemma:betweennotequal
   RAEAG  lemma:ray4
   MEABCD assumption
    NORAEAG
   NOMEABCD reductio
  qedcase
  case 5:BEAEG
   MEABCD assumption
    NOBEAEG
   NOMEABCD reductio
  qedcase
  case 6:BEAGE
   BEEGA  axiom:betweennesssymmetry
   NEEA lemma:betweennotequal
   RAEAG  lemma:ray4
   MEABCD assumption
    NORAEAG
   NOMEABCD reductio
  qedcase
 NOMEABCD  cases
NOMEABCD  reductio
% That is Euclid's conclusion.

BEEHF  lemma:collinearbetween
COABC assumption
 EQCC cn:equalityreflexive
 COCDC defn:collinear
 MEABCD defn:meet
NCABC reductio
BEDFC axiom:betweennesssymmetry
CODCE  assumption
 CODFC defn:collinear
 COCDE lemma:collinearorder
 COCDF lemma:collinearorder
 NEDC lemma:betweennotequal
 NECD lemma:inequalitysymmetric
 CODEF lemma:collinear4
 COEFD lemma:collinearorder
 NCEFD
NCDCE reductio
ANBEESC+BEDHS postulate:Pasch-outer
BEESC
BEDHS
NEAD  lemma:betweennotequal
NEDA  lemma:inequalitysymmetric
ANBEDAP+EEAPAD  postulate:extension

BEHAS  assumption
 BECSE axiom:betweennesssymmetry
 COCEH  assumption
  COHAS defn:collinear
  COCSE defn:collinear
  COEHF  defn:collinear
  COCES  lemma:collinearorder
  NECE  lemma:betweennotequal
  COEHS lemma:collinear4
  CODHS defn:collinear
  COHSE lemma:collinearorder
  COHSD lemma:collinearorder
  NEHS lemma:betweennotequal
  COSED lemma:collinear4
  COSEC  lemma:collinearorder
  NESE lemma:betweennotequal
  COEDC lemma:collinear4
  CODFC  defn:collinear
  COCDE lemma:collinearorder
  COCDF lemma:collinearorder
  NEDC lemma:betweennotequal
  NECD  lemma:inequalitysymmetric
  CODEF lemma:collinear4
  COEFD lemma:collinearorder
  NCEFD
 NCCEH reductio
 ANBEHTE+BECAT postulate:Pasch-outer
 BEHTE
 BECAT
 EQAA cn:equalityreflexive
 COABA defn:collinear
 NCABC
 OSCABT defn:oppositeside
 OSTABC lemma:oppositesidesymmetric
 COABF  assumption
  COCFD  defn:collinear
  COCDF  lemma:collinearorder
  MEABCD  defn:meet
 NCABF reductio
 SSFFAB lemma:samesidereflexive
 COAEB  defn:collinear
 COABE  lemma:collinearorder
 BEEHF
 BEETH  axiom:betweennesssymmetry
 BEETF  lemma:3.6b
 RAEFT lemma:ray4
 SSFTAB lemma:sameside2
 OSFABC  lemma:planeseparation
 ANBEFRC+COABR+NCABF  defn:oppositeside
 BEFRC
 COABR
 COFRC  defn:collinear
 COFCR  lemma:collinearorder
 COCFD  defn:collinear
 COFCD  lemma:collinearorder
 NECF  lemma:betweennotequal
 NEFC  lemma:inequalitysymmetric
 COCRD lemma:collinear4
 COCDR lemma:collinearorder
 MEABCD defn:meet
NOBEHAS  reductio
BEDHS
BEDHA  axiom:betweennesssymmetry
RAHSA  defn:ray
ORBEHAS|EQSA|BEHSA lemma:ray1
cases BEHSA:BEHAS|EQSA|BEHSA
 case 1:BEHAS
  NOBEHSA assumption
   NOBEHAS
  BEHSA reductio
 qedcase
 case 2:EQSA
  BECSE  axiom:betweennesssymmetry
  BECAE  cn:equalitysub
  COCAE  defn:collinear
  COEAC  lemma:collinearorder
  COEAB  lemma:collinearorder
  NEAE  lemma:betweennotequal
  NEEA  lemma:inequalitysymmetric
  COACB lemma:collinear4
  COABC  lemma:collinearorder
  EQCC  cn:equalityreflexive
  COCDC defn:collinear
  NOBEHSA assumption
   MEABCD defn:meet
  BEHSA reductio
 qedcase
 case 3:BEHSA
 qedcase
BEHSA cases 
BEDAP
BEDHA
BEHAP  lemma:3.6a
BESAP  lemma:3.6a
BEPAS  axiom:betweennesssymmetry
BEESC
COECP  assumption
 BEPAH  axiom:betweennesssymmetry
 COECS  defn:collinear
 NEEC lemma:betweennotequal
 COCPS lemma:collinear4
 CODAP defn:collinear
 COPAS defn:collinear
 COPSC lemma:collinearorder
 COPSA lemma:collinearorder
 NEPS lemma:betweennotequal
 COSCA lemma:collinear4
 COHSA defn:collinear
 COSAC lemma:collinearorder
 COSAH lemma:collinearorder
 NEAS lemma:betweennotequal
 NESA lemma:inequalitysymmetric
 COACH lemma:collinear4
 CODHA defn:collinear
 COAHD lemma:collinearorder
 COAHC lemma:collinearorder
 NEHA lemma:betweennotequal
 NEAH lemma:inequalitysymmetric
 COHDC lemma:collinear4
 CODFC defn:collinear
 COCDH lemma:collinearorder
 COCDF lemma:collinearorder
 NEDC lemma:betweennotequal
 NECD lemma:inequalitysymmetric
 CODHF lemma:collinear4
 COEFH defn:collinear
 COHFE lemma:collinearorder
 COHFD lemma:collinearorder
 NEHF lemma:betweennotequal
 COFED lemma:collinear4
 COEFD lemma:collinearorder
 NCEFD
NCECP  reductio
ANBEPQC+BEEAQ  postulate:Pasch-outer
BEPQC
BEEAQ
BECQP  axiom:betweennesssymmetry
COEAQ  defn:collinear
EQAA  cn:equalityreflexive
COABA defn:collinear
COAEB defn:collinear
COEAB lemma:collinearorder
NEAE lemma:betweennotequal
NEEA lemma:inequalitysymmetric
COAQB lemma:collinear4
COABQ lemma:collinearorder
EQAA  cn:equalityreflexive
COABA defn:collinear
COABC assumption
 EQCC cn:equalityreflexive
 COCDC defn:collinear
 MEABCD defn:meet
NCABC reductio
COABD assumption
 EQDD cn:equalityreflexive
 COCDD defn:collinear
 MEABCD defn:meet
NCABD reductio
SSCDAB  defn:sameside
ANNEAB+NECD+NOMEABCD+SSCDAB
PRABCD  defn:parallel