
theorem
for R being Ring,
    S being R-homomorphic Ring,
    f being Homomorphism of R,S
holds f is onto implies R/(ker f), S are_isomorphic
proof
let R be Ring, S be R-homomorphic Ring, f be Homomorphism of R,S;
set B = R/(ker f), I = ker f, T = Image f;
assume AS: f is onto;
then A: rng f = the carrier of S by FUNCT_2:def 3;
B: rng canHom f = the carrier of Image f by FUNCT_2:def 3
               .= rng f by defim;
then reconsider g = canHom f as Function of B,S by FUNCT_2:6;
C1: now let x,y be Element of B;
    thus g.(x+y)
      = (canHom f).x + (canHom f).y by VECTSP_1:def 20
     .= ((the addF of S)||(rng f)).((canHom f).x,(canHom f).y) by defim
     .= g.x + g.y by A;
    end;
C2: now let x,y be Element of B;
    thus g.(x*y)
      = (canHom f).x * (canHom f).y by GROUP_6:def 6
     .= ((the multF of S)||(rng f)).((canHom f).x,(canHom f).y) by defim
     .= g.x * g.y by A;
    end;
g.(1.B) = (canHom f).(1_B)
       .= 1_(Image f) by GROUP_1:def 13 .= 1.S by defim; then
C: g is additive multiplicative unity-preserving by C1,C2;
rng g = the carrier of S by B,AS,FUNCT_2:def 3;
then g is onto by FUNCT_2:def 3;
hence R/(ker f), S are_isomorphic by C;
end;
