reserve            S for satisfying_CongruenceSymmetry
                         satisfying_CongruenceEquivalenceRelation
                         TarskiGeometryStruct,
         a,b,c,d,e,f for POINT of S;
reserve S for satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        q,a,b,c,a9,b9,c9,x1,x2 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve       S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve         S for satisfying_CongruenceIdentity
                      satisfying_SegmentConstruction
                      satisfying_BetweennessIdentity
                      satisfying_Pasch
                      TarskiGeometryStruct,
        a,b,c,d,e for POINT of S;
reserve       S for satisfying_Tarski-model
                    TarskiGeometryStruct,
      a,b,c,d,p for POINT of S;
reserve                   S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9 for POINT of S;

theorem Satz4p13:
  for S being satisfying_Tarski-model
              TarskiGeometryStruct
  for a,b,c,a9,b9,c9 being POINT of S st Collinear a,b,c &
  a,b,c cong a9,b9,c9 holds Collinear a9,b9,c9
  proof
    let S be satisfying_Tarski-model TarskiGeometryStruct;
    let a,b,c,a9,b9,c9 be POINT of S;
    assume
A1: Collinear a,b,c & a,b,c cong a9,b9,c9;
A4: between b,c,a implies between b9,c9,a9
    proof
      assume
A5:   between b,c,a;
      b,c,a cong b9,c9,a9 by A1,Lm4p13p1;
      hence between b9,c9,a9 by A5,Satz4p6;
    end;
    between c,a,b implies between c9,a9,b9
    proof
      assume
A8:   between c,a,b;
      b,c,a cong b9,c9,a9 by A1,Lm4p13p1;
      then c,a,b cong c9,a9,b9 by Lm4p13p1;
      hence between c9,a9,b9 by A8,Satz4p6;
    end;
    hence thesis by A1,A4,Satz4p6;
  end;
