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
  0.R in S implies x,y Fr_Eq S
  proof
    assume
A1: 0.R in S;
    reconsider s1 = 0.R as Element of R;
A2: (x`1 * y`2 - y`1 * x`2) * s1 = 0.R;
    reconsider s1 as Element of S by A1;
    thus thesis by A1,A2;
  end;
