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TPABcd
COcdC
NECd
NEcd defn:tarski_parallel
OREQcd|EQcC|EQdC|BEdcC|BEcdC|BEcCd defn:collinear
cases TPABCd:EQcd|EQcC|EQdC|BEdcC|BEcdC|BEcCd
 case 1:EQcd
  NOTPABCd assumption
  NEcd
  TPABCd reductio
 qedcase
 case 2:EQcC
  TPABCd cn:equalitysub
 qedcase
 case 3:EQdC
  NOTPABCd assumption
   EQCd lemma:equalitysymmetric
  TPABCd reductio
 qedcase
 case 4:BEdcC
  BECcd axiom:betweennesssymmetry
  TPABCd lemma:parallelcollinear1
 qedcase
 case 5:BEcdC
  BECdc   axiom:betweennesssymmetry
  TPABdc  lemma:tarskiparallelflip
  TPABCc  lemma:parallelcollinear1
  TPABcC  lemma:tarskiparallelflip
  TPABdC  lemma:parallelcollinear2
  TPABCd  lemma:tarskiparallelflip
 qedcase
 case 5:BEcCd
  TPABCd lemma:parallelcollinear2
 qedcase
TPABCd cases  
TPABCd

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