reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;
reserve X,X9,Y for BCI-algebra,
  H9 for SubAlgebra of X9,
  G for SubAlgebra of X,

  A9 for non empty Subset of X9,
  I for Ideal of X,
  CI,K for closed Ideal of X,
  x,y,a,b for Element of X,
  RI for I-congruence of X,I,
  RK for I-congruence of X,K;
reserve f for BCI-homomorphism of X,X9;
reserve g for BCI-homomorphism of X9,X;
reserve h for BCI-homomorphism of X9,Y;

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;
