
theorem Th29:
  for L being non empty Poset, k being Function of L,L, X being
Subset of L st k is kernel & ex_sup_of X,L & X c= rng k holds sup X = k.(sup X)
proof
  let L be non empty Poset, k be Function of L,L, X be Subset of L such that
A1: k is projection and
A2: k <= id(L) and
A3: ex_sup_of X,L and
A4: X c= rng k;
A5: k is monotone by A1;
A6: k is idempotent by A1;
  k.(sup X) is_>=_than X
  proof
    let x be Element of L;
    assume
A7: x in X;
    sup X is_>=_than X by A3,YELLOW_0:30;
    then sup X >= x by A7;
    then k.(sup X) >= k.x by A5;
    hence thesis by A4,A6,A7,Lm2;
  end;
  then
A8: k.(sup X) >= sup X by A3,YELLOW_0:30;
  id(L).(sup X) >= k.(sup X) by A2,YELLOW_2:9;
  then sup X >= k.(sup X);
  hence thesis by A8,ORDERS_2:2;
end;
