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PRABCD
PRABCE
CRADBC
NEAB defn:parallel
NOORCRAEBC|CRACEB  assumption 
 ANNOCRAEBC+NOCRACEB
 NOCRAEBC
 NOCRACEB
 PRABEC  lemma:parallelflip
 CRACEB lemma:crisscross
ORCRAEBC|CRACEB reductio
ANBEAMD+BEBMC  defn:cross
BEBMC
cases COCDE:CRAEBC|CRACEB
 case 1:CRAEBC
  COCDE lemma:Playfairhelper
 qedcase
 case 2:CRACEB
  ANBEAmC+BEEmB defn:cross
  BEAmC
  BEEmB
  NECE  defn:parallel
  NEEC  lemma:inequalitysymmetric
  ANBEECe+EECeEC  lemma:extension
  BEECe
  COECe  defn:collinear
  NECe  lemma:betweennotequal
  NEeC  lemma:inequalitysymmetric
  PRABEC lemma:parallelflip
  PRABeC lemma:collinearparallel
  PRABCe lemma:parallelflip
  NCABC  lemma:parallelNC
  NCBCA  lemma:NCorder
  ANBEBHm+BEAHM  postulate:Pasch-inner
  NCAEC  lemma:parallelNC
  COECe  defn:collinear
  COCEe  lemma:collinearorder
  NCCEA  lemma:NCorder
  EQEE   cn:equalityreflexive
  COCEE  defn:collinear
  NEEe  lemma:betweennotequal
  NCEeA  lemma:NChelper
  BEAmC
  BEECe
  ANBEAFe+BEEmF  postulate:Pasch-outer
  BEAFe
  BEEmF
  BEeFA  axiom:betweennesssymmetry
  COEmF  defn:collinear
  COEmB  defn:collinear
  COmEF  lemma:collinearorder
  COmEB  lemma:collinearorder
  NEEm   lemma:betweennotequal
  NEmE    lemma:inequalitysymmetric
  COEFB   lemma:collinear4
  COEBF  lemma:collinearorder
  NEeE  lemma:inequalitysymmetric
  BEeFA
  EQEE   cn:equalityreflexive
  COeEE  defn:collinear
  EQBB   cn:equalityreflexive
  COBBA  defn:collinear
  NEBA   lemma:inequalitysymmetric
  PRABEC
  PRABCE  lemma:parallelflip
  COECe
  COCEe  lemma:collinearorder 
  NEEe
  PRABeE  lemma:collinearparallel
  PReEAB  lemma:parallelsymmetric
  PReEBA  lemma:parallelflip
  NOMEeEBA  defn:parallel
  BEEFB  lemma:collinearbetween
  BEBFE  axiom:betweennesssymmetry
  BEeCE  axiom:betweennesssymmetry
  NCBEC  lemma:parallelNC
  NCECB  lemma:NCorder
  COECe  defn:collinear
  EQEE    cn:equalityreflexive
  COECE   defn:collinear
  NCEeB  lemma:NChelper
  NCBEe  lemma:NCorder
  ANBEBKC+BEeKF  postulate:Pasch-inner
  BEBKC
  BEeKF
  BEeFA  axiom:betweennesssymmetry
  BEeKA  lemma:3.6b
  BEAKe  axiom:betweennesssymmetry
  BEAKe
  BEBKC
  ANBEAKe+BEBKC
  CRAeBC defn:cross
  COCDe  lemma:Playfairhelper
  COeCD  lemma:collinearorder
  COeCE lemma:collinearorder
  NECe  lemma:betweennotequal
  NEeC  lemma:inequalitysymmetric
  COCDE  lemma:collinear4
 qedcase
COCDE cases