
theorem Th1:
  for T being Noetherian sup-Semilattice for I being Ideal of T
  holds ex_sup_of I, T & sup I in I
proof
  let T be Noetherian sup-Semilattice;
  let I be Ideal of T;
  consider a being Element of T such that
A1: a in I and
A2: for b being Element of T st b in I holds not a < b by Def2;
A3: I is_<=_than a
  proof
    let d be Element of T;
    assume d in I;
    then a"\/"d in I by A1,WAYBEL_0:40;
    then
A4: not a < a"\/"d by A2;
    a <= a"\/"d by YELLOW_0:22;
    then a = a"\/"d by A4,ORDERS_2:def 6;
    hence thesis by YELLOW_0:22;
  end;
  for c being Element of T st I is_<=_than c holds a <= c by A1;
  hence thesis by A1,A3,YELLOW_0:30;
end;
