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

theorem
  a,b <= c,d & c,d <= a,b implies a,b equiv c,d
  proof
    assume
A1: a,b <= c,d & c,d <= a,b;
    then consider y be POINT of S such that
A2:  between c,y,d & a,b equiv c,y;
    consider p be POINT of S such that
A3: between c,d,p & c,p equiv a,b by A1,Satz5p5;
    consider q be POINT of S such that
A4: between a,q,b & c,d equiv a,q by A1;
    consider r be POINT of S such that
A5: between d,c,r & c,r equiv a,b by GTARSKI1:def 8;
A6: c = y implies a,b equiv c,d
    proof
      assume
A7:   c = y;
      then a = b by A2,GTARSKI1:def 7;
      then a = q by A4,GTARSKI1:def 10;
      hence thesis by A7,A2,A4,GTARSKI1:def 7;
    end;
    c <> y implies a,b equiv c,d
    proof
      assume
A8:   c <> y;
A9:   between d,y,c & between p,d,c by A2,A3,Satz3p2;
A10:  r <> c
      proof
        assume r = c;
        then a = b by A5,Satz2p2,GTARSKI1:def 7;
        hence contradiction by A2,Satz2p2,A8,GTARSKI1:def 7;
      end;
A11:  between r,c,y
      proof
        between y,c,r by A5,A9,Satz3p6p1;
        hence thesis by Satz3p2;
      end;
A12:  c <> d by A2,A8,GTARSKI1:def 10;
A13:  between r,c,p
      proof
        between r,c,d by A5,Satz3p2;
        hence thesis by A3,A12,Satz3p7p2;
      end;
      c,y equiv a,b by A2,Satz2p2;
      then y = p by A3,A10,A11,A13,Satz2p12;
      then between d,p,c & between p,d,c by A2,A3,Satz3p2;
      then p = d by Satz3p4;
      hence thesis by A3,Satz2p2;
    end;
    hence thesis by A6;
  end;
