
theorem Fiu:
  for T being non empty TopRelStr,
      R being non empty RelStr st
        the RelStr of T = the RelStr of R holds
    LAp T = LAp 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
      (LAp T).x = (LAp R).x
    proof
      let x be Element of bool the carrier of R;
      reconsider xx = x as Subset of R;
A2:   (LAp R).xx = LAp xx by ROUGHS_2:def 10;
      reconsider y = x as Subset of T by A0;
      (LAp T).y = LAp y by ROUGHS_2:def 10;
      hence thesis by A2,A0;
    end;
    hence thesis by A0;
  end;
