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 Lm63:
   Non_ZeroDiv_Set(A) = [#]A \ {0.A} & Non_ZeroDiv_Set(A) is
   without_zero non empty multiplicatively-closed Subset of A
   proof
A1:  Non_ZeroDiv_Set(A) = [#]A \ {0.A} by Th4;
     0.A in [#]A & 0.A in {0.A} by TARSKI:def 1; then
     Non_ZeroDiv_Set(A) is without_zero by A1, XBOOLE_0:def 5;
     hence thesis by Th4;
   end;
