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;
reserve p for POINT of S;
reserve r for POINT of S;
reserve x,y for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct;
reserve p,q,r,s for POINT of S;
reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
  a,b,p,q for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct,
                  A,B for Subset of S,
        a,b,c,p,q,r,s for POINT of S;

theorem
  for S being satisfying_Tarski-model non empty TarskiGeometryStruct
  for a,b,x,y being POINT of S st a <> b & x <> y &
  Line(a,b) = Line(x,y) holds a,b equal_line x,y
  proof
    let S be satisfying_Tarski-model non empty TarskiGeometryStruct;
    let a,b,x,y be POINT of S;
    assume that
A1: a <> b and
A1A: x <> y and
A2: Line(a,b) = Line(x,y);
    for c be POINT of S holds c on_line a,b iff c on_line x,y
    proof
      let c be POINT of S;
      hereby
        assume c on_line a,b;
        then Collinear a,b,c;
        then c in Line(x,y) by A2;
        then ex z be POINT of S st c = z & Collinear x,y,z;
        hence c on_line x,y by A1A;
      end;
      assume c on_line x,y;
      then Collinear x,y,c;
      then c in Line(a,b) by A2;
      then ex z be POINT of S st c = z & Collinear a,b,z;
      hence c on_line a,b by A1;
    end;
    hence thesis by A1,A1A;
  end;
