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;
reserve S for satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9,p,q for POINT of S;

theorem Satz4p14:
  Collinear a,b,c & a,b equiv a9,b9 implies ex c9 being POINT of S st
  a,b,c cong a9,b9,c9
  proof
    assume
A1: Collinear a,b,c & a,b equiv a9,b9;
    then per cases;
    suppose
A2:   between a,b,c;
      consider c9 be POINT of S such that
A3:   between a9,b9,c9 & b9,c9 equiv b,c by GTARSKI1:def 8;
A4:   b,c equiv b9,c9 by A3,Satz2p2;
      then a,c equiv a9,c9 by A1,A3,A2,Satz2p11;
      hence thesis by A1,A4,GTARSKI1:def 3;
    end;
    suppose between c,a,b;
      then
A5:   between b,a,c by Satz3p2;
      consider c9 be POINT of S such that
A6:   between b9,a9,c9 & a9,c9 equiv a,c by GTARSKI1:def 8;
      b,a equiv a9,b9 by A1,Satz2p4;
      then
A7:   b,a equiv b9,a9 by Satz2p5;
      a,c equiv a9,c9 by A6,Satz2p2;
      then
      b,a,c cong b9,a9,c9 by A5,A6,A7,Satz2p11;
      hence thesis by Lm4p14p1;
    end;
    suppose between b,c,a;
      then ex y be POINT of S st between a9,y,b9 & a,c,b cong a9,y,b9
        by A1,Satz3p2,Satz4p5;
      hence thesis by Lm4p14p2;
    end;
  end;
