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 bijective & g=f" implies g is bijective
proof
  assume
A1: f is bijective & g=f";
  dom f = the carrier of X by FUNCT_2:def 1;
  then rng g = the carrier of X by A1,FUNCT_1:33;
  then
A2: g is onto by FUNCT_2:def 3;
  g is one-to-one by A1,FUNCT_1:40;
  hence thesis by A2;
end;
