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 Satz4p16:
  a,b,c,d FS a9,b9,c9,d9 & a <> b implies c,d equiv c9,d9
  proof
    assume
A1: a,b,c,d FS a9,b9,c9,d9 & a <> b;
    then Collinear a,b,c;
    then per cases;
    suppose
A2:   between a,b,c;
      then
A3:   between a9,b9,c9 by A1,Satz4p6;
A4:   a,b equiv a9,b9 & a,c equiv a9,c9 & b,c equiv b9,c9 by A1,GTARSKI1:def 3;
      S is satisfying_SST_A5;
      hence thesis by A1,A4,A2,A3;
    end;
    suppose
A5:   between b,c,a;
      b,c,a cong b9,c9,a9 by A1,Lm4p13p1;
      then b,c,a,d IFS b9,c9,a9,d9 by A5,Satz4p6,A1;
      hence thesis by Satz4p2;
    end;
    suppose between c,a,b;
      then
A6:   between b,a,c by Satz3p2;
A7:   b,a,c cong b9,a9,c9 by A1,Lm4p14p1;
A8:   between b9,a9,c9 by A6,A7,Satz4p6;
      S is satisfying_SST_A5;
      hence thesis by A1,A8,A6,A7;
    end;
  end;
