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 Th53:
  I = Ker f & f is onto implies X./.RI,X9 are_isomorphic
proof
  assume that
A1: I = Ker f and
A2: f is onto;
  consider h being BCI-homomorphism of X./.RI,X9 such that
A3: f=h*nat_hom(RI) and
A4: h is one-to-one by A1,Th50;
  for y being object st y in the carrier of X9 ex z being object st z in
  the carrier of X./.RI&h.z = y
  proof
    let y be object;
    assume y in the carrier of X9;
    then y in rng f by A2,FUNCT_2:def 3;
    then consider x being object such that
A5: x in dom f and
A6: y=f.x by FUNCT_1:def 3;
    take (nat_hom RI).x;
A7: dom(nat_hom(RI)) = the carrier of X by FUNCT_2:def 1;
    then (nat_hom RI).x in rng nat_hom RI by A5,FUNCT_1:def 3;
    hence thesis by A3,A5,A6,A7,FUNCT_1:13;
  end;
  then rng h = the carrier of X9 by FUNCT_2:10;
  then h is onto by FUNCT_2:def 3;
  hence thesis by A4;
end;
