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 Satz4p2:
  a,b,c,d IFS a9,b9,c9,d9 implies b,d equiv b9,d9
  proof
    assume
A1: a,b,c,d IFS a9,b9,c9,d9;
    per cases;
    suppose
A2:   a = c;
      now
        thus c = b by A1,A2,GTARSKI1:def 10;
        a9 = c9 by A1,A2,Satz2p2,GTARSKI1:def 7;
        hence c9 = b9 by A1,GTARSKI1:def 10;
      end;
      hence thesis by A1;
    end;
    suppose
A3:   a <> c;
      consider e be POINT of S such that
A4:   between a,c,e & c,e equiv a,c by GTARSKI1:def 8;
A5:   c <> e by A3,A4,Satz2p2,GTARSKI1:def 7;
      consider e9 be POINT of S such that
A6:   between a9,c9,e9 & c9,e9 equiv c,e by GTARSKI1:def 8;
A7:   c,e equiv c9,e9 by A6,Satz2p2;
      S is satisfying_SST_A5; then
A8:   e,d equiv e9,d9 by A3,A4,A6,A1,A7;
      c,b equiv b9,c9 by A1,Satz2p4; then
A9:   c,b equiv c9,b9 by Satz2p5;
      e9,c9 equiv c,e by A6,Satz2p4;
      then e9,c9 equiv e,c by Satz2p5; then
A17:  e,c equiv e9,c9 by Satz2p2;
A18:  between e,c,b
      proof
        between b,c,e & between a,b,e by A1,A4,Satz3p6p1,Satz3p6p2;
        hence thesis by Satz3p2;
      end;
A19:  between e9,c9,b9
      proof
        between b9,c9,e9 & between a9,b9,e9 by A1,A6,Satz3p6p1,Satz3p6p2;
        hence thesis by Satz3p2;
      end;
      S is satisfying_SST_A5;
      hence thesis by A19,A18,A8,A9,A17,A1,A5;
    end;
  end;
