
theorem
  for L being non empty Poset, p being Function of L,L st p is
projection for Lk being non empty Subset of L st Lk = {k where k is Element of
L: p.k <= k} holds (p is sups-preserving implies subrelstr Lk is
sups-inheriting & Image p is sups-inheriting) & (p is directed-sups-preserving
  implies subrelstr Lk is directed-sups-inheriting & Image p is
  directed-sups-inheriting)
proof
  let L be non empty Poset, p be Function of L,L;
  assume
A1: p is projection;
  then reconsider
  Lc = {c where c is Element of L: c <= p.c} as non empty Subset of
  L by Th43;
  let Lk be non empty Subset of L such that
A2: Lk = {k where k is Element of L: p.k <= k};
A3: p is monotone by A1;
  then
A4: subrelstr Lc is sups-inheriting by Th49;
A5: Lc = the carrier of subrelstr Lc by YELLOW_0:def 15;
A6: the carrier of Image p = rng p by YELLOW_0:def 15
    .= Lc /\ Lk by A1,A2,Th42;
  then
A7: the carrier of Image p c= Lk by XBOOLE_1:17;
A8: Lk = the carrier of subrelstr Lk by YELLOW_0:def 15;
A9: the carrier of Image p c= Lc by A6,XBOOLE_1:17;
  hereby
    assume
A10: p is sups-preserving;
    thus
A11: subrelstr Lk is sups-inheriting
    proof
      let X be Subset of subrelstr Lk;
      the carrier of subrelstr Lk is Subset of L by YELLOW_0:def 15;
      then reconsider X9 = X as Subset of L by XBOOLE_1:1;
      assume
A12:  ex_sup_of X,L;
A13:  sup X9 is_>=_than p.:X9
      proof
        let y be Element of L;
        assume y in p.:X9;
        then consider x being Element of L such that
A14:    x in X9 and
A15:    y = p.x by FUNCT_2:65;
        reconsider x as Element of L;
        x in Lk by A8,A14;
        then
A16:    ex x9 being Element of L st x9 = x & x9 >= p.x9 by A2;
        sup X9 is_>=_than X9 by A12,YELLOW_0:30;
        then sup X9 >= x by A14;
        hence thesis by A15,A16,ORDERS_2:3;
      end;
      p preserves_sup_of X9 by A10;
      then ex_sup_of p.:X,L & sup (p.:X9) = p.(sup X9) by A12;
      then sup X9 >= p.(sup X9) by A13,YELLOW_0:30;
      hence thesis by A2,A8;
    end;
    thus Image p is sups-inheriting
    proof
      let X be Subset of Image p such that
A17:  ex_sup_of X,L;
      X c= Lk by A7;
      then
A18:  "\/"(X,L) in the carrier of subrelstr Lk by A8,A11,A17;
      subrelstr Lc is sups-inheriting & X c= the carrier of subrelstr Lc
      by A3,A9,A5,Th49;
      then "\/"(X,L) in the carrier of subrelstr Lc by A17;
      hence thesis by A6,A5,A8,A18,XBOOLE_0:def 4;
    end;
  end;
  assume
A19: p is directed-sups-preserving;
  thus
A20: subrelstr Lk is directed-sups-inheriting
  proof
    let X be directed Subset of subrelstr Lk;
    assume X <> {};
    then reconsider X9 = X as non empty directed Subset of L by YELLOW_2:7;
    assume
A21: ex_sup_of X,L;
A22: sup X9 is_>=_than p.:X9
    proof
      let y be Element of L;
      assume y in p.:X9;
      then consider x being Element of L such that
A23:  x in X9 and
A24:  y = p.x by FUNCT_2:65;
      reconsider x as Element of L;
      x in Lk by A8,A23;
      then
A25:  ex x9 being Element of L st x9 = x & x9 >= p.x9 by A2;
      sup X9 is_>=_than X9 by A21,YELLOW_0:30;
      then sup X9 >= x by A23;
      hence thesis by A24,A25,ORDERS_2:3;
    end;
    p preserves_sup_of X9 by A19;
    then ex_sup_of p.:X,L & sup (p.:X9) = p.(sup X9) by A21;
    then sup X9 >= p.(sup X9) by A22,YELLOW_0:30;
    hence thesis by A2,A8;
  end;
  let X be directed Subset of Image p such that
A26: X <> {} and
A27: ex_sup_of X,L;
  the carrier of Image p c= the carrier of subrelstr Lk by A7,YELLOW_0:def 15;
  then X is directed Subset of subrelstr Lk by YELLOW_2:8;
  then
A28: "\/"(X,L) in the carrier of subrelstr Lk by A20,A26,A27;
  X c= the carrier of subrelstr Lc by A9,A5;
  then "\/"(X,L) in the carrier of subrelstr Lc by A27,A4;
  hence thesis by A6,A5,A8,A28,XBOOLE_0:def 4;
end;
