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 Th26:
  Class(EqRel(S),x) = Class(EqRel(S),y) iff x,y Fr_Eq S
   proof
     set E = EqRel(S);
     thus Class(E,x) = Class(E,y) implies x,y Fr_Eq S
     proof
       assume Class(E,x) = Class(E,y);
       then x in Class(E,y) by EQREL_1:23;
       hence thesis by Th25;
     end;
     assume x,y Fr_Eq S;
     then x in Class(E,y) by Th25;
     hence thesis by EQREL_1:23;
  end;
