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 Th15:
  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 X is_less_than a &
  for b being Element of L9 st X is_less_than b holds a [= b
  proof
    let X be Subset of L9;
    set a = "\/" (X,L);
    reconsider a9 = a as Element of L9 by A1;
    take a9;
    X is_less_than a by LATTICE3:def 21;
    hence X is_less_than a9 by Th13;
    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 X is_less_than b9;
    then X is_less_than b by Th13;
    then
A2: a [= b by LATTICE3:def 21;
    a9 "/\" b9 = a "/\" b by Th11
      .= a9 by A2,LATTICES:4;
    hence thesis by LATTICES:4;
  end;
  hence thesis;
end;
