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;
