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

theorem
  for S being Subset of TOP-REAL 2, f being Homeomorphism of TOP-REAL 2
  st S is Jordan holds f.:S is Jordan
proof
  let S be Subset of TOP-REAL 2, f be Homeomorphism of TOP-REAL 2;
  set s = the Element of S`;
  assume
A1: S` <> {};
  then s in S`;
  then reconsider s as Element of TOP-REAL 2;
  given A1,A2 being Subset of TOP-REAL 2 such that
A2: S` = A1 \/ A2 and
A3: A1 misses A2 and
A4: (Cl A1) \ A1 = (Cl A2) \ A2 and
A5: A1 is_a_component_of S` & A2 is_a_component_of S`;
A6: not s in S by A1,XBOOLE_0:def 5;
  hereby
    assume (f.:S)` = {};
    then f.:S = the carrier of TOP-REAL 2 by Th9;
    then ex s9 being Element of TOP-REAL 2 st s9 in S & f.s = f. s9 by
FUNCT_2:65;
    hence contradiction by A6,FUNCT_2:56;
  end;
  take B1 = f.:A1, B2 = f.:A2;
  f.:(A1 \/ A2) = B1 \/ B2 by RELAT_1:120;
  hence (f.:S)` = B1 \/ B2 by A2,JORDAN1K:5;
  thus B1 misses B2 by A3,FUNCT_1:66;
  thus (Cl B1) \ B1 = (f.:Cl A1)\B1 by TOPS_2:60
    .= f.:((Cl A2) \ A2) by A4,FUNCT_1:64
    .= (f.:Cl A2) \ B2 by FUNCT_1:64
    .= (Cl B2) \ B2 by TOPS_2:60;
  f.:(S`) = (f.:S)` by JORDAN1K:5;
  hence thesis by A5,Th15;
end;
