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
  for a being Element of X st a is minimal holds f.a is minimal
proof
  let a be Element of X;
  assume a is minimal;
  then f.a=f.(a``)by BCIALG_2:29;
  then f.a=f.0.X\(f.(0.X\a)) by Def6;
  then f.a=f.0.X\(f.0.X\f.a) by Def6;
  then f.a=(f.0.X\f.a)` by Th35;
  then f.a=(f.a)`` by Th35;
  hence thesis by BCIALG_2:29;
end;
