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);

theorem Th31:
   0.R in S iff S~R is degenerated
   proof
A1:  S~R is degenerated implies 0.R in S
     proof
       assume S~R is degenerated; then
       Class(EqRel(S),1.(R,S)) = 0.(S~R) by Def6
       .= Class(EqRel(S),0.(R,S)) by Def6; then
       1.(R,S), 0.(R,S) Fr_Eq S by Th26; then
       consider s1 be Element of R such that
A3:    s1 in S and
A4:    (1.(R,S)`1 * 0.(R,S)`2 - 0.(R,S)`1 * 1.(R,S)`2) * s1 = 0.R;
       thus thesis by A3,A4;
     end;
     0.R in S implies S~R is degenerated
     proof
       assume 0.R in S; then
A6:    1.(R,S), 0.(R,S) Fr_Eq S;
       1.(S~R) = Class(EqRel(S),1.(R,S)) by Def6
       .= Class(EqRel(S),0.(R,S)) by A6,Th26 .= 0.(S~R) by Def6;
       hence thesis;
     end;
     hence thesis by A1;
   end;
