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 Th8:
  a '<' b implies All(a,A,G) '<' Ex(b,A,G)
proof
  assume a '<' b;
  then
A1: All(a,A,G) '<' All(b,A,G) by PARTIT_2:12;
  All(b,A,G) '<' Ex(b,A,G) by BVFUNC_4:17;
  hence thesis by A1,BVFUNC_1:15;
end;
