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 closed iff f"P is closed
proof
  let S, T be non empty TopStruct, f be Function of S, T;
  hereby
    assume
A1: f is being_homeomorphism;
    hence
A2: dom f = [#]S & rng f = [#]T & f is one-to-one;
    let P be Subset of T;
    hereby
      assume
A3:   P is closed;
      f is continuous by A1;
      hence f"P is closed by A3;
    end;
    assume f"P is closed;
    then f.:(f"P) is closed by A1,TOPS_2:58;
    hence P is closed by A2,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 closed iff f"P is closed;
  thus dom f = [#]S & rng f = [#]T & f is one-to-one by A4,A5,A6;
  thus f is continuous
  by A7;
  let R be Subset of S such that
A8: R is closed;
  for x1, x2 being Element of S st x1 in R & f.x1 = f.x2 holds x2 in R by A4,A6
;
  then
A9: f"(f.:R) = R by T_0TOPSP:1;
  (f/")"R = f.:R by A5,A6,TOPS_2:54;
  hence thesis by A7,A8,A9;
end;
