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