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 Th16:
  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 c9 is_less_than A9;
    then c is_less_than A9 by Th12;
    then
A3: c [= a by LATTICE3:34;
    c9 "/\" a9 = c "/\" a by Th11
      .= c9 by A3,LATTICES:4;
    hence c9 [= a9 by LATTICES:4;
  end;
  a is_less_than A9 by LATTICE3:34;
  then a9 is_less_than A9 by Th12;
  hence thesis by A2,LATTICE3:34;
end;
