
theorem Th50:
  for L be lower-bounded sup-Semilattice for S be non empty full
  SubRelStr of L st Bottom L in the carrier of S & the carrier of S is
join-closed Subset of L for x be Element of L holds waybelow x /\ (the carrier
  of S) is Ideal of S
proof
  let L be lower-bounded sup-Semilattice;
  let S be non empty full SubRelStr of L;
  assume that
A1: Bottom L in the carrier of S and
A2: the carrier of S is join-closed Subset of L;
  let x be Element of L;
  Bottom L << x by WAYBEL_3:4;
  then Bottom L in waybelow x by WAYBEL_3:7;
  then reconsider
  X = waybelow x /\ (the carrier of S) as non empty Subset of S by A1,
XBOOLE_0:def 4,XBOOLE_1:17;
  reconsider S1 = the carrier of S as join-closed Subset of L by A2;
A3: now
    let a,b be Element of S;
    reconsider a1 = a, b1 = b as Element of L by YELLOW_0:58;
    assume that
A4: a in X and
A5: b <= a;
    a in waybelow x by A4,XBOOLE_0:def 4;
    then
A6: a1 << x by WAYBEL_3:7;
    b1 <= a1 by A5,YELLOW_0:59;
    then b1 << x by A6,WAYBEL_3:2;
    then b in waybelow x by WAYBEL_3:7;
    hence b in X by XBOOLE_0:def 4;
  end;
  waybelow x /\ S1 is join-closed by Th33;
  hence thesis by A3,WAYBEL10:23,WAYBEL_0:def 19;
end;
