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;
reserve p for Element of Spectrum A;
reserve a,m,n for Element of A~p;
reserve f for Function of A,B;
reserve x for object;
reserve A for domRing;

theorem Th64:
   for a be Element of A holds a in Non_ZeroDiv_Set(A) iff a <> 0.A
   proof
     let a be Element of A;
     thus a in Non_ZeroDiv_Set(A) implies a <> 0.A
     proof
       assume a in Non_ZeroDiv_Set(A); then
       a in [#]A \ {0.A} by Lm63; then
       a in [#]A & not a in {0.A} by XBOOLE_0:def 5;
       hence thesis by TARSKI:def 1;
     end;
     assume a <> 0.A; then
     not a in {0.A} by TARSKI:def 1; then
     a in [#]A \ {0.A} by XBOOLE_0:def 5;
     hence thesis by Lm63;
   end;
