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
  for Z being SubAlgebra of X9 st the carrier of Z = rng f holds f is
  BCI-homomorphism of X,Z
proof
  let Z be SubAlgebra of X9;
A1: dom f = the carrier of X by FUNCT_2:def 1;
  assume the carrier of Z = rng f;
  then reconsider f9 = f as Function of X,Z by A1,RELSET_1:4;
  now
    let a,b;
    thus f9.a \f9.b = f.a \ f.b by Th34
      .= f9.(a \ b) by Def6;
  end;
  hence thesis by Def6;
end;
