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 Th42:
  f is onto implies for c being Element of X9 ex x st c = f.x
proof
A1: for c being set holds c in rng f iff ex x st c = f.x
  proof
    let c be set;
    thus c in rng f implies ex x st c = f.x
    proof
      assume c in rng f;
      then consider y being object such that
A2:   y in dom f and
A3:   f.y = c by FUNCT_1:def 3;
      reconsider y as Element of X by A2;
      take y;
      thus thesis by A3;
    end;
    given x such that
A4: c = f.x;
    the carrier of X = dom f by FUNCT_2:def 1;
    hence thesis by A4,FUNCT_1:def 3;
  end;
  assume f is onto;
  then rng f = the carrier of X9 by FUNCT_2:def 3;
  hence thesis by A1;
end;
