theorem
  for Z being SubAlgebra of X9 st the carrier of Z = rng f holds f is
  BCI-homomorphism of X,Z
proof
  let Z be SubAlgebra of X9;
A1: dom f = the carrier of X by FUNCT_2:def 1;
  assume the carrier of Z = rng f;
  then reconsider f9 = f as Function of X,Z by A1,RELSET_1:4;
  now
    let a,b;
    thus f9.a \f9.b = f.a \ f.b by Th34
      .= f9.(a \ b) by Def6;
  end;
  hence thesis by Def6;
end;
