
theorem ker0:
for R being Ring,
    S being R-homomorphic Ring,
    f being Homomorphism of R,S
holds f is monomorphism iff ker f = {0.R}
proof
let R be Ring, S be R-homomorphic Ring,
    f be Homomorphism of R,S;
A: now assume B: f is monomorphism;
   for x be object holds x in ker f iff x = 0.R
     proof
     let x be object;
     C: now assume AS: x in ker f;
        then reconsider a = x as Element of R;
        f.a = 0.S by AS,ker1 .= f.(0.R) by hom1;
        hence x = 0.R by B,FUNCT_2:19;
        end;
     now assume AS: x = 0.R;
       then reconsider a = x as Element of R;
       f.a = 0.S by AS,hom1;
       hence x in ker f;
       end;
     hence x in ker f iff x = 0.R by C;
     end;
   hence ker f = {0.R} by TARSKI:def 1;
   end;
now assume AS: ker f = {0.R};
   now let x,y be object;
     assume AS1: x in the carrier of R & y in the carrier of R & f.x = f.y;
     then reconsider a = x, b = y as Element of R;
     0.S = f.a - f.b by AS1,RLVECT_1:15
        .= f.(a-b) by hom4;
     then a - b in ker f;
     then 0.R = a + -b by AS,TARSKI:def 1;
     then a = -(-b) by RLVECT_1:6
           .= b;
     hence x = y;
     end;
   then f is one-to-one by FUNCT_2:19;
   hence f is monomorphism;
   end;
hence thesis by A;
end;
