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 Th25:
  x in Class(EqRel(S),y) iff x,y Fr_Eq S
  proof
    set E = EqRel(S);
    hereby
      assume x in Class(E,y);
      then [x,y] in E by EQREL_1:19;
      hence x,y Fr_Eq S by Def5;
    end;
    assume x,y Fr_Eq S;
    then [x,y] in E by Def5;
    hence thesis by EQREL_1:19;
  end;
