
theorem Th13:
  for X be set for c be Function of BoolePoset X,BoolePoset X st c
  is idempotent directed-sups-preserving holds inclusion c is
  directed-sups-preserving
proof
  let X be set;
  let c be Function of BoolePoset X,BoolePoset X;
  assume
A1: c is idempotent directed-sups-preserving;
  now
    let Y be Ideal of Image c;
    now
      "\/"(Y,BoolePoset X) in the carrier of BoolePoset X;
      then "\/"(Y,BoolePoset X) in dom c by FUNCT_2:def 1;
      then
A2:   c.("\/"(Y,BoolePoset X)) in rng c by FUNCT_1:def 3;
      Y c= the carrier of Image c;
      then
A3:   Y c= rng c by YELLOW_0:def 15;
      reconsider Y9 = Y as non empty directed Subset of BoolePoset X by
YELLOW_2:7;
      assume ex_sup_of Y,Image c;
A4:   ex_sup_of Y,BoolePoset X by YELLOW_0:17;
      c preserves_sup_of Y9 by A1,WAYBEL_0:def 37;
      then "\/"(c.:Y,BoolePoset X) in rng c by A4,A2,WAYBEL_0:def 31;
      then "\/"(Y,BoolePoset X) in rng c by A1,A3,YELLOW_2:20;
      then
A5:   "\/"(Y,BoolePoset X) in the carrier of Image c by YELLOW_0:def 15;
      thus ex_sup_of (inclusion c).:Y,BoolePoset X by YELLOW_0:17;
      thus sup ((inclusion c).:Y) = "\/"(Y,BoolePoset X) by FUNCT_1:92
        .= sup Y by A4,A5,YELLOW_0:64
        .= (inclusion c).sup Y;
    end;
    hence inclusion c preserves_sup_of Y by WAYBEL_0:def 31;
  end;
  hence thesis by WAYBEL_0:73;
end;
