reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem
  for S, T being non empty TopStruct, f being Function of S, T holds f
is being_homeomorphism iff dom f = [#]S & rng f = [#]T & f is one-to-one & for
  P being Subset of T holds P is open iff f"P is open
proof
  let S, T be non empty TopStruct, f be Function of S, T;
A1: [#]T <> {};
  hereby
    assume
A2: f is being_homeomorphism;
    hence
A3: dom f = [#]S & rng f = [#]T & f is one-to-one;
    let P be Subset of T;
    thus P is open implies f"P is open by A2,TOPS_2:43;
    assume f"P is open;
    then f.:(f"P) is open by A2,Th24;
    hence P is open by A3,FUNCT_1:77;
  end;
  assume that
A4: dom f = [#]S and
A5: rng f = [#]T and
A6: f is one-to-one and
A7: for P being Subset of T holds P is open iff f"P is open;
A8: now
    let R be Subset of S such that
A9: R is open;
    for x1, x2 being Element of S st x1 in R & f.x1 = f.x2 holds x2 in R
    by A4,A6;
    then
A10: f"(f.:R) = R by T_0TOPSP:1;
    (f/")"R = f.:R by A5,A6,TOPS_2:54;
    hence (f/")"R is open by A7,A9,A10;
  end;
  thus dom f = [#]S & rng f = [#]T & f is one-to-one by A4,A5,A6;
  thus f is continuous by A1,A7,TOPS_2:43;
  [#]S <> {};
  hence thesis by A8,TOPS_2:43;
end;
