 reserve L for Lattice;
 reserve I,P for non empty ClosedSubset of L;
reserve L for lower-bounded pseudocomplemented Lattice;
reserve L for Stone Lattice;
 reserve L1, L2 for Lattice;
 reserve p1, q1 for Element of L1;
 reserve p2, q2 for Element of L2;
 reserve L1, L2 for non empty Lattice;
reserve B for Boolean Lattice;

theorem PseudoInSquared:
  for L being Lattice, x1,x2 being Element of B, x being Element of L st
    L = B squared-latt & x = [x1,x2] holds x* = [x2`,x2`]
  proof
    let L be Lattice, x1,x2 be Element of B, x be Element of L;
    assume
A0:   L = B squared-latt & x = [x1,x2];
      x in the carrier of L; then
      x in B squared by SquaredCarrier,A0; then
      consider xx1,xx2 being Element of B such that
W1:   x = [xx1,xx2] & xx1 [= xx2;
aa:   xx1 = x1 & xx2 = x2 by W1,A0,XTUPLE_0:1;
      [x2`,x2`] in B squared; then
      reconsider y = [x2`,x2`] as Element of L by A0,SquaredCarrier;
Z1:   x "/\" y = [x1,x2] "/\" [x2`,x2`] by A0,MSUALG_7:11
         .= [x1 "/\" x2`, x2 "/\" x2`] by FILTER_1:35
         .= [x1 "/\" x2`, Bottom B] by LATTICES:20;
      x2` [= x1` by aa,W1,LATTICES:26; then
      x1 "/\" x2` [= x1 "/\" x1` by FILTER_0:5; then
      x1 "/\" x2` [= Bottom B by LATTICES:20; then
tt:   x1 "/\" x2` = Bottom B by BOOLEALG:9;
      for w being Element of L st x "/\" w = Bottom L holds w [= y
      proof
        let w be Element of L;
        assume
O1:     x "/\" w = Bottom L;
        w in the carrier of L; then
        w in B squared by A0,SquaredCarrier; then
        consider w1,w2 being Element of B such that
Y1:     w = [w1,w2] & w1 [= w2;
        [x1,x2] "/\" [w1,w2] = Bottom L by O1,Y1,A0,MSUALG_7:11; then
        [x1,x2] "/\" [w1,w2] = [Bottom B, Bottom B]
          by A0,SquaredBottom; then
        [x1 "/\" w1,x2 "/\" w2] = [Bottom B, Bottom B]
          by FILTER_1:35; then
        x1 "/\" w1 = Bottom B & x2 "/\" w2 = Bottom B by XTUPLE_0:1; then
Y2:     w2 [= x2` by LATTICES:25; then
        w1 [= x2` by Y1,LATTICES:7; then
        [w1,w2] "/\" [x2`,x2`] = [w1,w2] by LATTICES:4,Y2,FILTER_1:36; then
        w "/\" y = w by Y1,MSUALG_7:11,A0;
        hence thesis by LATTICES:4;
      end; then
      y is_a_pseudocomplement_of x by tt,Z1,A0,SquaredBottom;
      hence thesis by def3,A0;
  end;
