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 Th23:
   x,y Fr_Eq S implies y,x Fr_Eq S
   proof
     assume x,y Fr_Eq S; then
     consider s1 being Element of R such that
A2:  s1 in S and
A3:  (x`1 * y`2 - y`1 * x`2) * s1 = 0.R;
     reconsider w= y`1*x`2, v = x`1*y`2 as Element of R;
     (w-v)*s1=(-(v-w))*s1 by VECTSP_1:17
     .= -0.R by A3,VECTSP_1:9 .= 0.R;
     hence thesis by A2;
   end;
