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 Th29:
  G is independent implies All(Ex('not' a,A,G),B,G) '<' 'not' All(
  All(a,B,G),A,G)
proof
  Ex('not' a,A,G) = 'not' All(a,A,G) by BVFUNC_2:18;
  then
A1: All(Ex('not' a,A,G),B,G) = 'not' Ex(All(a,A,G),B,G) by BVFUNC_2:19;
  assume G is independent;
  hence thesis by A1,Th9,PARTIT_2:11;
end;
