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;
