
theorem Th33:
  for L being non empty Poset, f being Function of L,L st f is
  idempotent holds (corestr f)*(inclusion f) = id(Image f)
proof
  let L be non empty Poset, f be Function of L,L;
  assume
A1: f is idempotent;
  for s being Element of Image f holds ((corestr f)*(inclusion f)).s = s
  proof
    let s be Element of Image f;
    the carrier of Image f = rng corestr f by FUNCT_2:def 3;
    then consider l being object such that
A2: l in the carrier of L and
A3: (corestr f).l = s by FUNCT_2:11;
    reconsider l as Element of L by A2;
A4: (corestr f).l = f.l by Th30;
    thus ((corestr f)*(inclusion f)).s = (corestr f).((inclusion f).s) by
FUNCT_2:15
      .= (corestr f).s
      .= f.(f.l) by A3,A4,Th30
      .= s by A1,A3,A4,YELLOW_2:18;
  end;
  hence thesis by FUNCT_2:124;
end;
