
theorem Th42:
  for X being non empty set for L being up-complete non empty
  Poset holds L|^X is up-complete
proof
  let X be non empty set;
  let L be up-complete non empty Poset;
  now
    let A be non empty directed Subset of L|^X;
    reconsider B = A as non empty directed Subset of product (X --> L);
    now
      let x be Element of X;
      pi(B,x) is directed non empty by Th34;
      hence ex_sup_of pi(A,x), (X --> L).x by WAYBEL_0:75;
    end;
    hence ex_sup_of A,L|^X by Th30;
  end;
  hence thesis by WAYBEL_0:75;
end;
