reserve i for Integer,
  a, b, r, s for Real;

theorem
  for S being TopSpace, T being non empty TopSpace holds S,T
  are_homeomorphic iff the TopStruct of S, the TopStruct of T are_homeomorphic
proof
  let S be TopSpace, T be non empty TopSpace;
  set SS = the TopStruct of S;
  set TT = the TopStruct of T;
A1: [#]S = [#]SS & [#]T = [#]TT;
  thus S,T are_homeomorphic implies the TopStruct of S, the TopStruct of T
  are_homeomorphic
  proof
    given f being Function of S,T such that
A2: f is being_homeomorphism;
    reconsider g = f as Function of SS,TT;
A3: now
      let P be Subset of SS;
      reconsider R = P as Subset of S;
      thus g.:(Cl P) = f.:(Cl R) by TOPS_3:80
        .= Cl(f.:R) by A2,TOPS_2:60
        .= Cl(g.:P) by TOPS_3:80;
    end;
    take g;
    dom f = [#]S & rng f = [#]T by A2,TOPS_2:60;
    hence thesis by A1,A2,A3,TOPS_2:60;
  end;
  given f being Function of SS,TT such that
A4: f is being_homeomorphism;
  reconsider g = f as Function of S,T;
A5: now
    let P be Subset of S;
    reconsider R = P as Subset of SS;
    thus g.:(Cl P) = f.:(Cl R) by TOPS_3:80
      .= Cl(f.:R) by A4,TOPS_2:60
      .= Cl(g.:P) by TOPS_3:80;
  end;
  take g;
  dom f = [#]SS & rng f = [#]TT by A4,TOPS_2:60;
  hence thesis by A1,A4,A5,TOPS_2:60;
end;
