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
  f is onto implies f.:CI is closed Ideal of X9
proof
  assume f is onto;
  then reconsider Kf = f.:CI as Ideal of X9 by Th47;
  now
    let x9 be Element of Kf;
    consider x being object such that
    x in dom f and
A1: x in CI and
A2: x9 = f.x by FUNCT_1:def 6;
    reconsider x as Element of CI by A1;
    x` in the carrier of X;
    then x` in dom f by FUNCT_2:def 1;
    then x` in CI & [x`,f.(x`)] in f by BCIALG_1:def 19,FUNCT_1:1;
    then f.(x`)in f.:CI by RELAT_1:def 13;
    then f.(0.X)\f.x in f.:CI by Def6;
    hence x9` in Kf by A2,Th35;
  end;
  hence thesis by BCIALG_1:def 19;
end;
