 reserve L for Lattice;
 reserve I,P for non empty ClosedSubset of L;

theorem ThB:
  L is lower-bounded & Bottom L in I implies latt (L,I) is lower-bounded
    & Bottom latt (L,I) = Bottom L
  proof
    set b9 = the Element of latt (L,I);
    reconsider b = b9 as Element of L by FILTER_2:68;
    assume
A0: L is lower-bounded & Bottom L in I;
    set c = Bottom L;
    reconsider c9 = c as Element of latt (L,I) by FILTER_2:72,A0;
    ex c9 being Element of latt (L,I) st
    for a9 being Element of latt (L,I) holds c9 "/\" a9 = c9 & a9 "/\" c9 = c9
    proof
    take c9;
    let a9 be Element of latt (L,I);
    reconsider a = a9 as Element of L by FILTER_2:68;
    thus c9 "/\" a9 = c "/\" a by FILTER_2:73 .= c9 by A0;
    hence thesis;
    end;
    hence
W1: latt (L,I) is lower-bounded by LATTICES:def 13;
    for a9 being Element of latt (L,I) holds c9 "/\" a9 = c9 & a9 "/\" c9 = c9
    proof
      let a9 be Element of latt (L,I);
      reconsider a = a9 as Element of L by FILTER_2:68;
      thus c9 "/\" a9 = c "/\" a by FILTER_2:73  .= c9 by A0;
      hence thesis;
    end;
    hence thesis by LATTICES:def 16,W1;
  end;
