reserve R,R1 for commutative Ring;
reserve A,B for non degenerated commutative Ring;
reserve o,o1,o2 for object;
reserve r,r1,r2 for Element of R;
reserve a,a1,a2,b,b1 for Element of A;
reserve f for Function of R, R1;
reserve p for Element of Spectrum A;
reserve S for non empty multiplicatively-closed Subset of R;
reserve u,v,w,x,y,z for Element of Frac(S);
reserve a, b, c for Element of Frac(S);
reserve x, y, z for Element of S~R;
reserve S for without_zero non empty multiplicatively-closed Subset of A;

theorem Lm50:
   not 0.A in S implies ker canHom(S) c= ZeroDiv_Set(A)
   proof
     assume
A1:  not 0.A in S;
     for o st o in ker canHom(S) holds o in ZeroDiv_Set(A)
     proof
       let o;
       assume o in ker canHom(S); then
       consider v1 be Element of A such that
A3:    v1 = o and
A4:    (canHom(S)).v1 = 0.(S~A);
       1.A in S by C0SP1:def 4; then
       reconsider w = [v1,1.A] as Element of Frac(S) by Def3;
       Class(EqRel(S),0.(A,S)) = (canHom(S)).v1 by A4,Def6
       .= Class(EqRel(S),(frac1(S)).v1) by Def7
       .= Class(EqRel(S),w) by Def4; then
       0.(A,S), w Fr_Eq S by Th26; then
       consider t1 be Element of A such that
A5:    t1 in S and
A6:    (0.(A,S)`1 * w`2 - w`1* 0.(A,S)`2)*t1 = 0.A;
A7:    0.A = ((- 1.A) * v1)*t1 by A6,VECTSP_2:29
       .= (- 1.A) * (v1*t1) by GROUP_1:def 3
       .= - (v1*t1) by VECTSP_2:29;
A8:    0.A = -(-v1*t1) by A7 .= v1*t1;
       v1 is zero_divisible by A1,A5,A8;
       hence thesis by A3;
     end;
     hence thesis;
   end;
