
theorem Th47:
  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} for pc being Function of subrelstr Lc,subrelstr Lc st pc = p|Lc
  holds pc is closure
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is idempotent and
A2: p is monotone;
  let Lc be non empty Subset of L such that
A3: Lc = {c where c is Element of L: c <= p.c};
  let pc be Function of subrelstr Lc,subrelstr Lc such that
A4: pc = p|Lc;
A5: dom pc = the carrier of subrelstr Lc by FUNCT_2:def 1;
  hereby
    now
      let x be Element of subrelstr Lc;
A6:   x is Element of L by YELLOW_0:58;
A7:   pc.x = p.x by A4,A5,FUNCT_1:47;
      then p.(p.x) = pc.(pc.x) by A4,A5,FUNCT_1:47
        .= (pc*pc).x by A5,FUNCT_1:13;
      hence (pc*pc).x = pc.x by A1,A7,A6,YELLOW_2:18;
    end;
    hence pc*pc = pc by FUNCT_2:63;
    thus pc is monotone
    proof
      let x1,x2 be Element of subrelstr Lc;
      reconsider x19 = x1, x29 = x2 as Element of L by YELLOW_0:58;
      assume x1 <= x2;
      then x19 <= x29 by YELLOW_0:59;
      then
A8:   p.x19 <= p.x29 by A2;
      pc.x1 = p.x19 & pc.x2 = p.x29 by A4,A5,FUNCT_1:47;
      hence thesis by A8,YELLOW_0:60;
    end;
  end;
  now
    let x be Element of subrelstr Lc;
    reconsider x9=x as Element of L by YELLOW_0:58;
    x in the carrier of subrelstr Lc;
    then x in Lc by YELLOW_0:def 15;
    then
A9: ex c being Element of L st x = c & c <= p.c by A3;
    pc.x = p.x9 by A4,A5,FUNCT_1:47;
    then x <= pc.x by A9,YELLOW_0:60;
    hence (id subrelstr Lc).x <= pc.x;
  end;
  hence thesis by YELLOW_2:9;
end;
