 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;

theorem ProductPCompl:
  L1 is pseudocomplemented 01_Lattice & L2 is pseudocomplemented 01_Lattice
  implies [p1,p2]* = [p1*,p2*]
  proof
    assume
A1: L1 is pseudocomplemented 01_Lattice &
    L2 is pseudocomplemented 01_Lattice;
A3: p1* is_a_pseudocomplement_of p1 by def3,A1;
    p2* is_a_pseudocomplement_of p2 by def3,A1; then
    [p1*,p2*] is_a_pseudocomplement_of [p1,p2] by A3,ProductPsCompl,A1;
    hence thesis by def3,A1;
  end;
