theorem
  for D being complete /\-distributive Lattice, a being Element of D holds
  a "\/" "/\"(X,D) = "/\"({a"\/"b1 where b1 is Element of D: b1 in X}, D) &
  "/\"(X,D) "\/" a = "/\"({b2"\/" a where b2 is Element of D: b2 in X}, D)
proof
  let D be complete /\-distributive Lattice, a be Element of D;
  defpred X[set] means $1 in X;
A1: "/\"({a"\/"b where b is Element of D: b in X}, D) [= a "\/" "/\"(X,D)
  by Th36;
A2: a"\/""/\"(X,D) [= "/\"({a"\/" b where b is Element of D: b in X}, D) by
Th35;
  hence a"\/""/\"(X,D) = "/\"({a"\/"b where b is Element of D: b in X}, D)
  by A1,LATTICES:8;
  deffunc U(Element of D) = $1"\/"a;
  deffunc V(Element of D) = a"\/"$1;
A3: for b being Element of D holds V(b) = U(b);
  {V(b) where b is Element of D: X[b]} = {U(c) where c is Element of D: X[c]}
  from FRAENKEL:sch 5(A3);
  hence thesis by A1,A2,LATTICES:8;
end;
