
theorem Th13:
  for T being up-complete non empty Poset for S being non empty
  Poset, f being Function of T,T st f is_a_retraction_of T,S holds f is
  directed-sups-preserving projection
proof
  let T be up-complete non empty Poset;
  let S be non empty Poset, f be Function of T,T;
  assume
A1: f is_a_retraction_of T,S;
  then reconsider g = f as directed-sups-preserving Function of T,S;
  f is idempotent by A1,Th11;
  then
A2: f = f*f by QUANTAL1:def 9
    .= (f|rng f)*f by FUNCT_4:2
    .= (f|the carrier of S)*f by A1,Th8,Th10
    .= (id S)*f by A1
    .= (id the carrier of S)*g;
A3: S is full directed-sups-inheriting non empty SubRelStr of T by A1;
  then
A4: incl(S,T) is directed-sups-preserving by WAYBEL21:10;
  the carrier of S c= the carrier of T by A3,YELLOW_0:def 13;
  then
A5: incl(S,T) = id the carrier of S by YELLOW_9:def 1;
  hence f is directed-sups-preserving by A2,A4,WAYBEL20:28;
  f is directed-sups-preserving idempotent Function of T,T by A1,A2,A4,A5,Th11,
WAYBEL20:28;
  hence thesis;
end;
