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
  G is independent implies Ex(Ex('not' a,A,G),B,G) '<' Ex('not' All(a,B,
  G),A,G)
proof
A1: 'not' All(All(a,A,G),B,G) = Ex(Ex('not' a,A,G),B,G) by Th21;
  assume G is independent;
  hence thesis by A1,Th15;
end;
