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;

theorem Th15:
  Frac(S) = [:[#]R,S:]
   proof
     o in Frac(S) implies o in [:[#]R,S:]
     proof
       assume o in Frac(S); then
       consider a,b be Element of R such that
A2:    o = [a,b] and
A3:    b in S by Def3;
       thus thesis by A2,A3,ZFMISC_1:def 2;
     end; then
A3:  Frac(S) c= [:[#]R,S:];
     o in [:[#]R,S:] implies o in Frac(S)
     proof
       assume o in [:[#]R,S:]; then
       consider o1,o2 be object such that
A5:    o1 in [#]R and
A6:    o2 in S and
A7:    o = [o1,o2] by ZFMISC_1:def 2;
       consider a,b be Element of R such that
A8:    a = o1 and
A9:    b = o2 by A5,A6;
       o = [a,b] by A8,A9,A7;
       hence thesis by A6,A9,Def3;
     end; then
     [:[#]R,S:] c= Frac(S);
     hence thesis by A3,XBOOLE_0:def 10;
   end;
