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(I_el(Y),PA,G) = I_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(I_el Y,PA,G).z=TRUE
  proof
    let z be Element of Y;
    z in EqClass(z,CompF(PA,G)) & (I_el Y).z=TRUE by BVFUNC_1:def 11
,EQREL_1:def 6;
    hence thesis by BVFUNC_1:def 17;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
