reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,i for set;
reserve r,r1,r2 for Real;

theorem Th14:
  for L be complete Lattice, L9 be SubLattice of L st L9 is
  /\-inheriting holds L9 is complete
proof
  let L be complete Lattice;
  let L9 be SubLattice of L such that
A1: L9 is /\-inheriting;
  for X being Subset of L9 ex a being Element of L9 st a is_less_than X &
  for b being Element of L9 st b is_less_than X holds b [= a
  proof
    let X be Subset of L9;
    set a = "/\" (X,L);
    reconsider a9 = a as Element of L9 by A1;
    take a9;
    a is_less_than X by LATTICE3:34;
    hence a9 is_less_than X by Th12;
    let b9 be Element of L9;
    the carrier of L9 c= the carrier of L by NAT_LAT:def 12;
    then reconsider b = b9 as Element of L;
    assume b9 is_less_than X;
    then b is_less_than X by Th12;
    then
A2: b [= a by LATTICE3:39;
    b9 "/\" a9 = b "/\" a by Th11
      .= b9 by A2,LATTICES:4;
    hence thesis by LATTICES:4;
  end;
  hence thesis by VECTSP_8:def 6;
end;
