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 Thequiv2:
  for S being satisfying_Tarski-model non empty TarskiGeometryStruct
  for a,b,x,y being POINT of S st
  a,b equal_line x,y holds Line(a,b) = Line(x,y)
  proof
    let S be satisfying_Tarski-model non empty TarskiGeometryStruct;
    let a,b,x,y be POINT of S;
    assume
A1: a,b equal_line x,y;
    Line(a,b) = Line(x,y)
    proof
A2:   Line(a,b) c= Line(x,y)
      proof
        let z be object;
        assume z in Line(a,b);
        then consider z9 be POINT of S such that
A3:     z = z9 and
A4:     Collinear a,b,z9;
        z9 on_line x,y by A1,A4,Thequiv1;
        then Collinear x,y,z9;
        hence z in Line(x,y) by A3;
      end;
      Line(x,y) c= Line(a,b)
      proof
        let z be object;
        assume z in Line(x,y);
        then consider z9 be POINT of S such that
A5:     z = z9 and
A6:     Collinear x,y,z9;
        z9 on_line a,b by A6,A1,Thequiv1;
        then Collinear a,b,z9;
        hence thesis by A5;
      end;
      hence thesis by A2;
    end;
    hence thesis;
  end;
