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);
reserve a, b, c for Element of Frac(S);
reserve x, y, z for Element of S~R;

theorem Th35:
  x = Class(EqRel(S),a) & y = Class(EqRel(S),b) implies
    x+y = Class(EqRel(S),a+b)
   proof
     consider a1, b1 being Element of Frac(S) such that
A1:  x = Class(EqRel(S),a1) & y = Class(EqRel(S),b1) and
A2:  (the addF of S~R).(x,y) = Class(EqRel(S),a1+b1) by Def6;
     assume x = Class(EqRel(S),a) & y = Class(EqRel(S),b); then
     a,a1 Fr_Eq S & b,b1 Fr_Eq S by A1,Th26;
     hence thesis by A2,Th26,Th28;
   end;
