reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem Th9:
  for S,T being non empty TopSpace, S0 being non empty SubSpace of
  S, T0 being non empty SubSpace of T, f being Function of S,T st f is
being_homeomorphism for g being Function of S0,T0 st g = f|S0 & g is onto holds
  g is being_homeomorphism
proof
  let S,T be non empty TopSpace, S0 be non empty SubSpace of S, T0 be non
  empty SubSpace of T, f be Function of S,T such that
A1: f is being_homeomorphism;
A2: rng f = [#]T by A1;
A3: f" is continuous by A1;
  let g be Function of S0,T0 such that
A4: g = f|S0 and
A5: g is onto;
A6: g = f|the carrier of S0 by A4,TMAP_1:def 4;
  then
A7: f.:the carrier of S0 = rng g by RELAT_1:115
    .= the carrier of T0 by A5,FUNCT_2:def 3;
  thus dom g = [#]S0 by FUNCT_2:def 1;
  thus
 rng g = [#]T0 by A5,FUNCT_2:def 3;
A8: f is one-to-one by A1;
  hence
A9: g is one-to-one by A6,FUNCT_1:52;
A10:  f is onto by A2,FUNCT_2:def 3;
  f is continuous by A1;
  then g is continuous Function of S0,T by A4;
  hence g is continuous by Th8;
  g" = (f qua Function|the carrier of S0)" by A5,A6,A9,TOPS_2:def 4
    .= (f qua Function)"|(f.:the carrier of S0) by A8,RFUNCT_2:17
    .= f"|(the carrier of T0) by A10,A8,A7,TOPS_2:def 4
    .= f"|T0 by TMAP_1:def 4;
  then g" is continuous Function of T0,S by A3;
  hence thesis by Th8;
end;
