
theorem Th7:
  for S,T being non empty Poset, f being Function st f
  is_a_retraction_of T,S holds f*incl(S,T) = id S
proof
  let S,T be non empty Poset, f be Function such that
  f is directed-sups-preserving Function of T,S and
A1: f|the carrier of S = id S and
A2: S is directed-sups-inheriting full SubRelStr of T;
  the carrier of S c= the carrier of T by A2,YELLOW_0:def 13;
  hence f*incl(S, T) = f*id the carrier of S by YELLOW_9:def 1
    .= id S by A1,RELAT_1:65;
end;
