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
  h*f is BCI-homomorphism of X,Y
proof
  reconsider g = h*f as Function of X,Y;
  now
    let a,b;
    thus g.(a \ b) = h.(f.(a \ b)) by FUNCT_2:15
      .= h.(f.a \ f.b) by Def6
      .= (h.(f.a)) \ (h.(f.b)) by Def6
      .= (g.a)\(h.(f.b)) by FUNCT_2:15
      .= g.a \ g.b by FUNCT_2:15;
  end;
  hence thesis by Def6;
end;
