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 Satz6p19:
  a <> b & A is_line & a in A & b in A & B is_line & a in B & b in B
  implies A = B
  proof
    assume that
A1: a <> b and
A2: A is_line and
A3: a in A and
A4: b in A and
A5: B is_line and
A6: a in B and
A7: b in B;
    consider pa,qa be POINT of S such that pa <> qa and
A8: A = Line(pa,qa) by A2;
    consider pb,qb be POINT of S such that pb <> qb and
A9: B = Line(pb,qb) by A5;
    Line(pa,qa) = Line(a,b) & Line(pb,qb) = Line(a,b)
      by A1,A3,A4,A8,A2,A6,A7,A9,A5,Satz6p18;
    hence thesis by A8,A9;
  end;
