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 All(All(a,A,G),B,G) '<' All(Ex(a,B,G),A,G)
proof
  assume G is independent;
  then
A1: All(All(a,A,G),B,G) = All(All(a,B,G),A,G) by PARTIT_2:15;
  All(a,B,G) '<' Ex(a,B,G) by Th8;
  hence thesis by A1,PARTIT_2:12;
end;
