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 Satz5p5:
  a,b <= c,d iff
  (ex x being POINT of S st between a,b,x & a,x equiv c,d)
  proof
A1: a,b <= c,d implies (ex x be POINT of S st between a,b,x & a,x equiv c,d)
    proof
      assume a,b <= c,d;
      then consider y be POINT of S such that
A2:   between c,y,d & a,b equiv c,y;
      Collinear c,y,d by A2;
      then consider x be POINT of S such that
A3:   c,y,d cong a,b,x by A2,Satz2p2,Satz4p14;
      a,x equiv c,d by A3,Satz2p2;
      hence thesis by A2,A3,Satz4p6;
    end;
    (ex x be POINT of S st between a,b,x & a,x equiv c,d) implies a,b <= c,d
    proof
      assume ex x be POINT of S st between a,b,x & a,x equiv c,d;
      then consider x be POINT of S such that
A4:   between a,b,x & a,x equiv c,d;
A5:   Collinear x,a,b by A4;
      x,a equiv c,d by A4,Satz2p4;
      then consider y be POINT of S such that
A6:   x,a,b cong d,c,y by A5,Satz2p5,Satz4p14;
      a,b,x cong c,y,d
      proof
A9:     a,x equiv c,d
        proof
          a,x equiv d,c by A6,Satz2p4;
          hence thesis by Satz2p5;
        end;
        b,x equiv y,d
        proof
          b,x equiv d,y by A6,Satz2p4;
          hence thesis by Satz2p5;
        end;
        hence thesis by A6,A9;
      end;
      hence thesis by A4,Satz4p6;
    end;
    hence thesis by A1;
  end;
