reserve p1, p2 for Point of TOP-REAL 2,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2;

theorem Th11:
  for T1,T2 being non empty TopSpace, f being Function of T1,T2 st
  f is being_homeomorphism for A being Subset of T1 st A is a_component
  holds f.:A is a_component
proof
  let T1,T2 be non empty TopSpace, f be Function of T1,T2;
  assume
A1: f is being_homeomorphism;
  let A be Subset of T1;
  assume that
A2: A is connected and
A3: for B being Subset of T1 st B is connected holds A c= B implies A = B;
  thus f.:A is connected by A1,A2,TOPS_2:61;
  let B be Subset of T2;
  rng f = the carrier of T2 by A1;
  then
A4: f.:(f"B) = B by FUNCT_1:77;
A5: f"(f.:A) = A by A1,FUNCT_1:94;
  assume that
A6: B is connected and
A7: f.:A c= B;
  f"B is connected by A1,A6,TOPS_2:62;
  hence thesis by A3,A4,A5,A7,RELAT_1:143;
end;
