reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;
reserve p9,q9 for Element of LattPOSet L;
reserve C for complete Lattice,
  a,a9,b,b9,c,d for Element of C,
  X,Y for set;

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;
A1: "\/"({a"/\"b where b is Element of D: b in X}, D) [= a "/\" "\/"(X,D)
  by Th32;
A2: a"/\""\/"(X,D) [= "\/"({a"/\" b where b is Element of D: b in X}, D) by
Th33;
  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;
  defpred X[set] means $1 in X;
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;
