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 Th33:
   x = Class(EqRel(S),a) & y = Class(EqRel(S),b) implies
     x*y = Class(EqRel(S),a*b)
   proof
     assume that
A1:  x = Class(EqRel(S),a) and
A2:  y = Class(EqRel(S),b);
     consider a1, b1 being Element of Frac(S) such that
A3:  x = Class(EqRel(S),a1) and
A4:  y = Class(EqRel(S),b1) and
A5:  (the multF of S~R).(x,y) = Class(EqRel(S),a1*b1) by Def6;
A6:  a1,a Fr_Eq S by A1,A3,Th26;
     b1,b Fr_Eq S by A2,A4,Th26;
     hence thesis by A5,Th26,A6,Th27;
   end;
