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 Th30:
  for s be Element of S holds
    x=[s,s] implies x,1.(R,S) Fr_Eq S
   proof
     let s be Element of S;
     assume
A1:  x=[s,s];
     reconsider s1 = 1.R as Element of R;
A2:  (x`1 * 1.(R,S)`2 - 1.(R,S)`1 * x`2)*s1 = 0.R by A1,RLVECT_1:5;
     s1 in S by C0SP1:def 4;
     hence thesis by A2;
   end;
