
theorem Th43:
  for T being non empty TopSpace, S being non empty SubSpace of T
  for f being Function of T,S st f is being_a_retraction holds rng f = the
  carrier of S
proof
  let T be non empty TopSpace, S be non empty SubSpace of T;
  let f be Function of T,S such that
A1: for W being Point of T st W in the carrier of S holds f.W=W;
  thus rng f c= the carrier of S;
  let x be object;
A2: [#]S = the carrier of S;
  [#]T = the carrier of T;
  then
A3: the carrier of S c= the carrier of T by A2,PRE_TOPC:def 4;
  assume
A4: x in the carrier of S;
  then x in the carrier of T by A3;
  then
A5: x in dom f by FUNCT_2:def 1;
  f.x = x by A1,A3,A4;
  hence thesis by A5,FUNCT_1:def 3;
end;
