reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;
reserve S for non empty TopStruct;
reserve f for Function of T,S;
reserve SS for non empty TopSpace;
reserve f for Function of TS,SS;

theorem
  TS is compact & SS is T_2 & rng f = [#]SS & f is
  one-to-one & f is continuous implies f is being_homeomorphism
proof
  assume that
A1: TS is compact and
A2: SS is T_2 and
A3: rng f = [#] SS and
A4: f is one-to-one and
A5: f is continuous;
A6: dom f = [#]TS by FUNCT_2:def 1;
  for P being Subset of TS holds P is closed iff f.:P is closed
  proof
    let P be Subset of TS;
A7: P c= f"(f.:P) by A6,FUNCT_1:76;
    thus P is closed implies f.:P is closed by A1,A2,A3,A5,Th16;
    assume f.:P is closed;
    then
A8: f"(f.:P) is closed by A5;
    f"(f.:P) c= P by A4,FUNCT_1:82;
    hence thesis by A8,A7,XBOOLE_0:def 10;
  end;
  hence thesis by A6,A3,A4,TOPS_2:58;
end;
