
theorem Th54:
  for T being non empty TopSpace for S being non empty SubSpace of
T, f being continuous Function of T,S st f is being_a_retraction holds f*incl(S
  ,T) = id S
proof
  let T be non empty TopSpace, S be non empty SubSpace of T;
  let f be continuous Function of T,S such that
A1: f is being_a_retraction;
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;
  then
A4: incl(S,T) = id the carrier of S by YELLOW_9:def 1;
  now
    let x be Element of S;
    reconsider y = x as Point of T by A3;
    thus (f*incl(S,T)).x = f.((incl(S,T)).x) by FUNCT_2:15
      .= f.x by A4
      .= y by A1
      .= (id S).x;
  end;
  hence thesis by FUNCT_2:63;
end;
