
theorem ThA:
  for L being distributive Lattice, S being Sublattice of L holds
    S is distributive
  proof
    let L be distributive Lattice, S be Sublattice of L;
    let a,b,c be Element of S;
    reconsider aa = a, bb = b, cc = c as Element of L by FILTER_2:68;
A2: b "\/" c = bb "\/" cc by MSUALG_7:11;
    reconsider ab = aa "/\" bb, ac = aa "/\" cc as Element of L;
A4: ab = a "/\" b & ac = a "/\" c by MSUALG_7:11;
    a "/\" (b "\/" c) = aa "/\" (bb "\/" cc) by MSUALG_7:11,A2
       .= (aa "/\" bb) "\/" (aa "/\" cc) by LATTICES:def 11
       .= (a "/\" b) "\/" (a "/\" c) by MSUALG_7:11,A4;
    hence thesis;
  end;
