reserve n for Element of NAT,
  V for Subset of TOP-REAL n,
  s,s1,s2,t,t1,t2 for Point of TOP-REAL n,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  a,p ,p1,p2,q,q1,q2 for Point of TOP-REAL 2;

theorem
  for S,T being non empty TopStruct, f being Function of S,T, A being
  Subset of T st f is being_homeomorphism & A is compact holds f"A is compact
proof
  let S,T be non empty TopStruct, f be Function of S,T, A be Subset of T such
  that
A1: f is being_homeomorphism and
A2: A is compact;
A3: rng f = [#]T & f is one-to-one by A1,TOPS_2:def 5;
  f" is being_homeomorphism by A1,TOPS_2:56;
  then
A4: rng (f") = [#]S by TOPS_2:def 5;
  f" is continuous by A1,TOPS_2:def 5;
  then f".:A is compact by A2,A4,COMPTS_1:15;
  hence thesis by A3,TOPS_2:55;
end;
