 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;

theorem ProductIsSStone:
  L1 is satisfying_Stone_identity pseudocomplemented 01_Lattice &
  L2 is satisfying_Stone_identity pseudocomplemented 01_Lattice implies
  [:L1,L2:] is satisfying_Stone_identity
  proof
    assume that
A1: L1 is satisfying_Stone_identity pseudocomplemented 01_Lattice and
A2: L2 is satisfying_Stone_identity pseudocomplemented 01_Lattice;
    set L = [:L1,L2:];
    for x being Element of L holds x* "\/" (x**) = Top L
    proof
      let x be Element of L;
      consider x1,x2 being object such that
A3:   x1 in the carrier of L1 & x2 in the carrier of L2 & x = [x1,x2]
        by ZFMISC_1:def 2;
      reconsider x1 as Element of L1 by A3;
      reconsider x2 as Element of L2 by A3;
X1:   [x1*,x2*] = x* by ProductPCompl,A3,A1,A2;
A4:   x1* "\/" (x1**) = Top L1 by A1,SatStone;
      [x1* "\/" (x1**), x2* "\/" (x2**)] = [Top L1, Top L2]
        by A4,A2,SatStone
           .= Top L by FILTER_1:43,A1,A2; then
      Top L = [x1*,x2*] "\/" [x1**,x2**] by FILTER_1:35
           .= x* "\/" (x**) by X1,ProductPCompl,A1,A2;
      hence thesis;
    end;
    hence thesis;
  end;
