reserve Y for non empty set,
  a, b for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  A, B for a_partition of Y;

theorem Th40:
  'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G)
proof
  All('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) & 'not' Ex(Ex(a,A
  ,G), B,G) = All('not' Ex(a,A,G),B,G) by Th28,BVFUNC_2:19;
  hence thesis by BVFUNC_2:18;
end;
