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 Th42:
  G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' All('not'
  All(a,B,G),A,G)
proof
  assume G is independent;
  then 'not' Ex(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) by Th33;
  hence thesis by BVFUNC_2:19;
end;
