theorem
  nat_hom RI is onto
proof
  set f = nat_hom RI;
  set Y = X./.RI;
  reconsider Y as BCI-algebra;
  reconsider f as BCI-homomorphism of X,Y;
  for y being object st y in the carrier of Y ex x being object st x in the
  carrier of X & y = f.x
  proof
    let y be object;
    assume
A1: y in the carrier of Y;
    then reconsider y as Element of Y;
    consider x being object such that
A2: x in the carrier of X and
A3: y = Class(RI,x) by A1,EQREL_1:def 3;
    take x;
    thus thesis by A2,A3,Def10;
  end;
  then rng f = the carrier of Y by FUNCT_2:10;
  hence thesis by FUNCT_2:def 3;
end;
