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

theorem
  for S, T being non empty TopSpace, f being Function of S,T st f is
  one-to-one onto continuous open holds f is being_homeomorphism
proof
  let S, T be non empty TopSpace, f be Function of S,T such that
A1: f is one-to-one and
A2: f is onto and
A3: f is continuous and
A4: f is open;
A5: [#]T <> {};
A6: dom f = the carrier of S by FUNCT_2:def 1;
A7: for P being Subset of S holds P is open iff f.:P is open
  proof
    let P be Subset of S;
    thus P is open implies f.:P is open by A4;
    assume f.:P is open;
    then f"(f.:P) is open by A3,A5,TOPS_2:43;
    hence thesis by A1,A6,FUNCT_1:94;
  end;
  dom f = [#]S & rng f = [#]T by A2,FUNCT_2:def 1;
  hence thesis by A1,A7,TOPGRP_1:25;
end;
