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(O_el(Y),PA,G) = O_el(Y)
proof
  let a be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  for z being Element of Y holds All(O_el Y,PA,G).z=FALSE
  proof
    let z be Element of Y;
    z in EqClass(z,CompF(PA,G)) & (O_el Y).z=FALSE by BVFUNC_1:def 10
,EQREL_1:def 6;
    hence thesis by BVFUNC_1:def 16;
  end;
  hence thesis by BVFUNC_1:def 10;
end;
