reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN, G being Subset of PARTITIONS(
  Y), PA,PB being a_partition of Y holds All(a,PA,G) '<' Ex(a,PB,G)
proof
  let a be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA,PB be a_partition of Y;
  All(a,PA,G) '<' a & a '<' Ex(a,PB,G) by BVFUNC_2:11,12;
  hence thesis by BVFUNC_1:15;
end;
