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 one-to-one iff Ker(f) = {0.X}
proof
  thus f is one-to-one implies Ker(f) = {0.X}
  proof
    assume
A1: f is one-to-one;
    thus Ker(f) c= {0.X}
    proof
      let a be object;
      assume a in Ker f;
      then
A2:   ex x being Element of X st a=x & f.x = 0.X9;
      then reconsider a as Element of X;
      f.a = f.(0.X) by A2,Th35;
      then a = 0.X by A1,FUNCT_2:19;
      hence thesis by TARSKI:def 1;
    end;
    let a be object;
    assume
A3: a in {0.X};
    then reconsider a as Element of X by TARSKI:def 1;
    a = 0.X by A3,TARSKI:def 1;
    then f.a = 0.X9 by Th35;
    hence thesis;
  end;
  assume
A4: Ker(f) = {0.X};
  now
    let a,b;
    assume that
A5: a <> b and
A6: f.a = f.b;
    f.b \ f.a = 0.X9 by A6,BCIALG_1:def 5;
    then f.(b\a) = 0.X9 by Def6;
    then b\a in Ker f;
    then
A7: b\a = 0.X by A4,TARSKI:def 1;
    f.a \ f.b = 0.X9 by A6,BCIALG_1:def 5;
    then f.(a \ b) = 0.X9 by Def6;
    then a\b in Ker f;
    then a\b =0.X by A4,TARSKI:def 1;
    hence contradiction by A5,A7,BCIALG_1:def 7;
  end;
  then for a,b st f.a = f.b holds a = b;
  hence thesis by GROUP_6:1;
end;
