
theorem
  for T being non empty TopRelStr,
      R being non empty RelStr st
        the RelStr of T = the RelStr of R holds
    UAp T = UAp R
  proof
    let T be non empty TopRelStr,
        R be non empty RelStr;
    assume
A0: the RelStr of T = the RelStr of R;
    for x being Element of bool the carrier of R holds
      (UAp T).x = (UAp R).x
    proof
      let x be Element of bool the carrier of R;
      reconsider xx = x as Subset of R;
A2:   (UAp R).xx = UAp xx by ROUGHS_2:def 11;
      reconsider y = x as Subset of T by A0;
      (UAp T).y = UAp y by ROUGHS_2:def 11;
      hence thesis by A2,A0;
    end;
    hence thesis by A0;
  end;
