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
  All(All('not' a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G)
proof
  'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) & All('not' Ex(a,A,G)
  ,B,G ) = All(All('not' a,A,G),B,G) by BVFUNC_2:19;
  hence thesis by Th43;
end;
