reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;
reserve T for non empty TopSpace,
  S for TopSpace,
  P1 for Subset of S,
  f for Function of T, S;
reserve T for TopSpace,
  S for non empty TopSpace,
  P for Subset of T,
  f for Function of T, S;

theorem :: JORDAN18:2, AK, 21.02.2006
  for S,T being non empty TopSpace, f being Function of S,T, A being
Subset of T st f is being_homeomorphism & A is connected holds f"A is connected
proof
  let S,T be non empty TopSpace, f be Function of S,T, A be Subset of T such
  that
A1: f is being_homeomorphism and
A2: A is connected;
  f" is continuous by A1;
  then
A3: f".:A is connected by A2,Th61;
  rng f = [#]T & f is one-to-one by A1;
  hence thesis by A3,Th55;
end;
