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 Th33:
  G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' 'not' Ex(
  All(a,B,G),A,G)
proof
  assume G is independent;
  then
A1: Ex(Ex(a,A,G),B,G) = Ex(Ex(a,B,G),A,G) by PARTIT_2:16;
  All(a,B,G) '<' Ex(a,B,G) by Th8;
  then Ex(All(a,B,G),A,G) '<' Ex(Ex(a,A,G),B,G) by A1,PARTIT_2:13;
  hence thesis by PARTIT_2:11;
end;
