
theorem Th33:
  for L being bounded distributive well-complemented
  preOrthoLattice, x, y being Element of L holds x "/\" y = (x` "\/" y`)`
proof
  let L be bounded distributive well-complemented preOrthoLattice;
  let x, y be Element of L;
A1: x` "\/" Top L = Top L;
A2: y` "\/" Top L = Top L;
A3: y` is_a_complement_of y by Def10;
  then
A4: y` "\/" y = Top L;
  (x "/\" y)` is_a_complement_of (x "/\" y) by Def10;
  then
A5: (x "/\" y)` "\/" (x "/\" y) = Top L & (x "/\" y)` "/\" (x "/\" y) =
  Bottom L;
A6: x` is_a_complement_of x by Def10;
  then
A7: x` "\/" x = Top L;
A8: y` "/\" y = Bottom L by A3;
A9: x` "/\" x = Bottom L by A6;
A10: (x` "\/" y`) "/\" (x "/\" y) = (x "/\" y "/\" x`) "\/" (x "/\" y "/\" y`
  ) by LATTICES:def 11
    .= (y "/\" (x "/\" x`)) "\/" (x "/\" y "/\" y`) by LATTICES:def 7
    .= (y "/\" Bottom L) "\/" (x "/\" (y "/\" y`)) by A9,LATTICES:def 7
    .= Bottom L "\/" (x "/\" Bottom L) by A8
    .= Bottom L "\/" Bottom L
    .= Bottom L;
  (x` "\/" y`) "\/" (x "/\" y) = (x` "\/" y` "\/" x) "/\" (x` "\/" y`
  "\/" y) by LATTICES:11
    .= (y` "\/" x` "\/" x) "/\" Top L by A4,A1,LATTICES:def 5
    .= Top L "/\" Top L by A7,A2,LATTICES:def 5
    .= Top L;
  then (x "/\" y)` = x` "\/" y` by A10,A5,LATTICES:12;
  hence thesis by Th32;
end;
