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 Th9:
  G is independent implies All(All(a,A,G),B,G) '<' Ex(All(a,B,G),A,G)
proof
  assume G is independent;
  then All(All(a,A,G),B,G) = All(All(a,B,G),A,G) by PARTIT_2:15;
  hence thesis by Th8;
end;
