 reserve L for AD_Lattice;
 reserve x,y,z for Element of L;
 reserve L for GAD_Lattice;
 reserve x,y,z for Element of L;
reserve L for with_zero GAD_Lattice,
        x,y for Element of L;

theorem :: Corollary 3.12
 for a being Element of L, S being non empty ClosedSubset of L,
     b being Element of latt(L,S) st b = a &
    S = the set of all x "/\" a where x is Element of L
  holds latt(L,S) is Lattice-like distributive &
  (for c being Element of latt(L,S) holds b "\/" c = b & c "\/" b = b & c [= b)
  proof
    let a be Element of L;
    let S be non empty ClosedSubset of L;
    let b be Element of latt(L,S);
    assume
Z5: b = a;
    assume
Z2: S = the set of all x "/\" a where x is Element of L;
    for y1,y2 being Element of latt(L,S) holds y1 "/\" y2 = y2 "/\" y1
    proof
      let y1,y2 be Element of latt(L,S);
      y1 in the carrier of latt(L,S);
      then y1 in S by Defx4;
      then consider x1 be Element of L such that
B1:   y1 = x1 "/\" a by Z2;
      y2 in the carrier of latt(L,S);
      then y2 in S by Defx4;
      then consider x2 be Element of L such that
B2:   y2 = x2 "/\" a by Z2;
      thus y1 "/\" y2 = x1 "/\" a "/\" (x2 "/\" a) by B1,B2,Thx4
      .= a "/\" x1 "/\" (x2 "/\" a) by Lem310
      .= a "/\" (x1 "/\" (x2 "/\" a)) by LATTICES:def 7
      .= a "/\" (x1 "/\" x2 "/\" a) by LATTICES:def 7
      .= a "/\" (x2 "/\" x1 "/\" a) by Lem310
      .= a "/\" (x2 "/\" (x1 "/\" a)) by LATTICES:def 7
      .= a "/\" x2 "/\" (x1 "/\" a) by LATTICES:def 7
      .= x2 "/\" a "/\" (x1 "/\" a) by Lem310
      .= y2 "/\" y1 by B1,B2,Thx4;
    end;
    then
Z12: latt(L,S) is meet-commutative;
    latt(L,S) is GAD_Lattice by Thx1;
    hence latt(L,S) is Lattice-like distributive by Z12,Th31141;
    let c be Element of latt(L,S);
    c in the carrier of latt(L,S);
    then c in S by Defx4;
    then consider x be Element of L such that
A1: c = x "/\" a by Z2;
    reconsider d = c as Element of L by A1;
    thus b "\/" c = a "\/" (x "/\" a) by A1,Z5,Thx3 .= b by Z5,ThA5;
    thus
A2: c "\/" b = (x "/\" a) "\/" a by A1,Z5,Thx3 .= b by Z5,LATTICES:def 8;
    thus c [= b by A2;
  end;
