
theorem
  for L being non empty Poset, p being Function of L,L st p is
projection for Lc being non empty Subset of L st Lc = {c where c is Element of
L: c <= p.c} holds (p is infs-preserving implies subrelstr Lc is
infs-inheriting & Image p is infs-inheriting) & (p is filtered-infs-preserving
  implies subrelstr Lc is filtered-infs-inheriting & Image p is
  filtered-infs-inheriting)
proof
  let L be non empty Poset, p be Function of L,L;
  assume
A1: p is projection;
  then reconsider
  Lk = {k where k is Element of L: p.k <= k} as non empty Subset of
  L by Th43;
  let Lc be non empty Subset of L such that
A2: Lc = {c where c is Element of L: c <= p.c};
A3: p is monotone by A1;
  then
A4: subrelstr Lk is infs-inheriting by Th50;
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 infs-preserving;
    thus
A11: subrelstr Lc is infs-inheriting
    proof
      let X be Subset of subrelstr Lc;
      the carrier of subrelstr Lc is Subset of L by YELLOW_0:def 15;
      then reconsider X9 = X as Subset of L by XBOOLE_1:1;
      assume
A12:  ex_inf_of X,L;
A13:  inf 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 Lc by A5,A14;
        then
A16:    ex x9 being Element of L st x9 = x & x9 <= p.x9 by A2;
        inf X9 is_<=_than X9 by A12,YELLOW_0:31;
        then inf X9 <= x by A14;
        hence inf X9 <= y by A15,A16,ORDERS_2:3;
      end;
      p preserves_inf_of X9 by A10;
      then ex_inf_of p.:X,L & inf (p.:X9) = p.(inf X9) by A12;
      then inf X9 <= p.(inf X9) by A13,YELLOW_0:31;
      hence thesis by A2,A5;
    end;
    thus Image p is infs-inheriting
    proof
      let X be Subset of Image p such that
A17:  ex_inf_of X,L;
      X c= Lc by A9;
      then
A18:  "/\"(X,L) in the carrier of subrelstr Lc by A5,A11,A17;
      subrelstr Lk is infs-inheriting & X c= the carrier of subrelstr Lk
      by A3,A7,A8,Th50;
      then "/\"(X,L) in the carrier of subrelstr Lk by A17;
      hence thesis by A6,A5,A8,A18,XBOOLE_0:def 4;
    end;
  end;
  assume
A19: p is filtered-infs-preserving;
  thus
A20: subrelstr Lc is filtered-infs-inheriting
  proof
    let X be filtered Subset of subrelstr Lc;
    assume X <> {};
    then reconsider X9 = X as non empty filtered Subset of L by YELLOW_2:7;
    assume
A21: ex_inf_of X,L;
A22: inf 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 Lc by A5,A23;
      then
A25:  ex x9 being Element of L st x9 = x & x9 <= p.x9 by A2;
      inf X9 is_<=_than X9 by A21,YELLOW_0:31;
      then inf X9 <= x by A23;
      hence inf X9 <= y by A24,A25,ORDERS_2:3;
    end;
    p preserves_inf_of X9 by A19;
    then ex_inf_of p.:X,L & inf (p.:X9) = p.(inf X9) by A21;
    then inf X9 <= p.(inf X9) by A22,YELLOW_0:31;
    hence thesis by A2,A5;
  end;
  let X be filtered Subset of Image p such that
A26: X <> {} and
A27: ex_inf_of X,L;
  the carrier of Image p c= the carrier of subrelstr Lc by A9,YELLOW_0:def 15;
  then X is filtered Subset of subrelstr Lc by YELLOW_2:8;
  then
A28: "/\"(X,L) in the carrier of subrelstr Lc by A20,A26,A27;
  X c= the carrier of subrelstr Lk by A7,A8;
  then "/\"(X,L) in the carrier of subrelstr Lk by A27,A4;
  hence thesis by A6,A5,A8,A28,XBOOLE_0:def 4;
end;
