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 Satz4p5:
  between a,b,c & a,c equiv a9,c9 implies ex b9 st between a9,b9,c9 &
  a,b,c cong a9,b9,c9
  proof
    assume
A1: between a,b,c & a,c equiv a9,c9;
    consider d9 be POINT of S such that
A2: between c9,a9,d9 & a9,d9 equiv c9,a9 by GTARSKI1:def 8;
    per cases;
    suppose a9 = d9;
      then c9 = a9 by A2,Satz2p2,GTARSKI1:def 7;
      then a = c by A1,GTARSKI1:def 7;
      then
A2BIS: a = b by A1,GTARSKI1:def 10;
      take a9;
      thus thesis by A2BIS,Satz2p8,A1,Satz3p1,Satz3p2;
    end;
    suppose
A2TER:  a9 <> d9;
      consider b9 be POINT of S such that
A3:   between d9,a9,b9 & a9,b9 equiv a,b by GTARSKI1:def 8;
      consider c99 be POINT of S such that
A4:   between d9,b9,c99 & b9,c99 equiv b,c by GTARSKI1:def 8;
A5:   between a9,b9,c99 & between d9,a9,c99 by A3,A4,Satz3p6p1,Satz3p6p2;
A6:   a,b equiv a9,b9 by A3,Satz2p2;
A7:   b,c equiv b9,c99 by A4,Satz2p2;
      then
A8:   a,c equiv a9,c99 by A1,A5,A6,Satz2p11;
A9:   c99 = c9
      proof
A10:    between d9,a9,c9 by A2,Satz3p2;
A11:    between d9,a9,c99 by A3,A4,Satz3p6p2;
A12:    a9,c9 equiv a,c by A1,Satz2p2;
        a9,c99 equiv a,c by A8,Satz2p2;
        hence thesis by A2TER,A10,A11,A12,Satz2p12;
      end;
      between a9,b9,c9 by A3,A4,Satz3p6p1,A9;
      hence thesis by A6,A7,A1,A9,GTARSKI1:def 3;
    end;
  end;
