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 Th17:
  for L be complete Lattice, L9 be SubLattice of L st L9 is
  \/-inheriting for A9 be Subset of L9 holds "\/" (A9,L) = "\/" (A9,L9)
proof
  let L be complete Lattice;
  let L9 be SubLattice of L;
  assume
A1: L9 is \/-inheriting;
  then reconsider L91 = L9 as complete SubLattice of L;
  let A9 be Subset of L9;
  set a = "\/" (A9,L);
  reconsider a9 = a as Element of L91 by A1;
A2: now
    let c9 be Element of L91;
    the carrier of L91 c= the carrier of L by NAT_LAT:def 12;
    then reconsider c = c9 as Element of L;
    assume A9 is_less_than c9;
    then A9 is_less_than c by Th13;
    then
A3: a [= c by LATTICE3:def 21;
    a9 "/\" c9 = a "/\" c by Th11
      .= a9 by A3,LATTICES:4;
    hence a9 [= c9 by LATTICES:4;
  end;
  A9 is_less_than a by LATTICE3:def 21;
  then A9 is_less_than a9 by Th13;
  hence thesis by A2,LATTICE3:def 21;
end;
