reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for a being Function of Y,BOOLEAN, PA being a_partition of Y
  holds All(a,PA,G) '<' a
proof
  let a be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  let z be Element of Y;
A1: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume All(a,PA,G).z= TRUE;
  hence a.z=TRUE by A1,BVFUNC_1:def 16;
end;
