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 Th45:
  A9 is Ideal of X9 implies f"A9 is Ideal of X
proof
  assume
A1: A9 is Ideal of X9;
A2: now
    let x,y be Element of X;
    assume that
A3: x\y in f"A9 and
A4: y in f"A9;
    f.(x\y) in A9 by A3,FUNCT_2:38;
    then
A5: f.x\f.y in A9 by Def6;
    f.y in A9 by A4,FUNCT_2:38;
    then f.x in A9 by A1,A5,BCIALG_1:def 18;
    hence x in f"A9 by FUNCT_2:38;
  end;
  0.X9 in A9 by A1,BCIALG_1:def 18;
  then f.0.X in A9 by Th35;
  then 0.X in f"A9 by FUNCT_2:38;
  hence thesis by A2,BCIALG_1:def 18;
end;
