
theorem
  for X, Y being non empty reflexive RelStr st [:X,Y:] is antisymmetric
  holds X is antisymmetric & Y is antisymmetric
proof
  let X, Y be non empty reflexive RelStr such that
A1: [:X,Y:] is antisymmetric;
  for x,y being Element of X st x <= y & y <= x holds x = y
  proof
    set z = the Element of Y;
A2: z <= z;
    let x, y be Element of X;
    assume x <= y & y <= x;
    then [x,z] <= [y,z] & [y,z] <= [x,z] by A2,Th11;
    then [x,z] = [y,z] by A1,YELLOW_0:def 3;
    hence thesis by XTUPLE_0:1;
  end;
  hence X is antisymmetric by YELLOW_0:def 3;
  for x,y being Element of Y st x <= y & y <= x holds x = y
  proof
    set z = the Element of X;
A3: z <= z;
    let x, y be Element of Y;
    assume x <= y & y <= x;
    then [z,x] <= [z,y] & [z,y] <= [z,x] by A3,Th11;
    then [z,x] = [z,y] by A1,YELLOW_0:def 3;
    hence thesis by XTUPLE_0:1;
  end;
  hence thesis by YELLOW_0:def 3;
end;
