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 Ex(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 Ex(O_el Y,PA,G).z=FALSE
  proof
    let z be Element of Y;
    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (O_el Y)
    .x<>TRUE by BVFUNC_1:def 10;
    hence thesis by BVFUNC_1:def 17;
  end;
  hence thesis by BVFUNC_1:def 10;
end;
