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
  Ker(nat_hom RK) = K
proof
  set h=nat_hom RK;
  set Y = X./.RK;
  thus Ker h c= K
  proof
    let y be object;
    assume y in Ker h;
    then consider x being Element of X such that
A1: y=x and
A2: h.x = 0.Y;
    Class(RK,0.X)=Class(RK,x) by A2,Def10;
    then x in Class(RK,0.X) by EQREL_1:23;
    then [0.X,x] in RK by EQREL_1:18;
    then x\0.X in K by BCIALG_2:def 12;
    hence thesis by A1,BCIALG_1:2;
  end;
  let y be object;
  assume
A3: y in K;
  then reconsider x=y as Element of X;
  x\0.X in K & x` in K by A3,BCIALG_1:2,def 19;
  then [0.X,x] in RK by BCIALG_2:def 12;
  then x in Class(RK,0.X) by EQREL_1:18;
  then Class(RK,0.X)=Class(RK,x) by EQREL_1:23;
  then h.x =0.Y by Def10;
  hence thesis;
end;
