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 Th41:
  Ker f is closed Ideal of X
proof
  now
    let x be object;
    assume x in Ker f;
    then ex x9 being Element of X st x=x9 & f.x9 = 0.X9;
    hence x in the carrier of X;
  end;
  then
A1: Ker f is non empty Subset of X by TARSKI:def 3;
A2: now
    let x,y;
    assume x\y in Ker f & y in Ker f;
    then
    (ex y9 being Element of X st y=y9 & f.y9 = 0.X9 )& ex x9 being Element
    of X st x\y=x9 & f.x9 = 0.X9;
    then f.x \ 0.X9 = 0.X9 by Def6;
    then f.x = 0.X9 by BCIALG_1:2;
    hence x in Ker f;
  end;
  f.(0.X) = 0.X9 by Th35;
  then 0.X in Ker f;
  then reconsider Kf=Ker f as Ideal of X by A1,A2,BCIALG_1:def 18;
  Kf is closed
  proof
    let x be Element of Kf;
    x in Kf;
    then
A3: ex x9 being Element of X st x=x9 & f.x9=0.X9;
    f.(x`)=f.(0.X)\f.x by Def6;
    then f.(x`)=(0.X9)` by A3,Th35;
    then f.(x`)=0.X9 by BCIALG_1:def 5;
    hence thesis;
  end;
  hence thesis;
end;
