
theorem
  for T being non empty TopSpace, S being non empty SubSpace of T for f
  being continuous Function of T,S st f is being_a_retraction holds f is
  idempotent
proof
  let T be non empty TopSpace, S be non empty SubSpace of T;
A1: [#]S = the carrier of S;
  let f be continuous Function of T,S such that
A2: f is being_a_retraction;
A3: rng f = the carrier of S by A2,Th43;
A4: dom f = the carrier of T by FUNCT_2:def 1;
  [#]T = the carrier of T;
  then
A5: the carrier of S c= the carrier of T by A1,PRE_TOPC:def 4;
A6: now
    let x be object;
    assume
A7: x in the carrier of T;
    then
A8: f.x in rng f by A4,FUNCT_1:def 3;
    (f*f).x = f.(f.x) by A4,A7,FUNCT_1:13;
    hence (f*f).x = f.x by A2,A5,A3,A8;
  end;
  dom (f*f) = the carrier of T by A5,A4,A3,RELAT_1:27;
  then f*f = f by A4,A6,FUNCT_1:2;
  hence thesis by QUANTAL1:def 9;
end;
