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 Th27:
  x,u Fr_Eq S & y,v Fr_Eq S implies Fracmult(x,y),Fracmult(u,v) Fr_Eq S
   proof
     assume that
A1:  x,u Fr_Eq S and
A2:  y,v Fr_Eq S;
     consider s1 being Element of R such that
A3:  s1 in S and
A4:  (x`1 * u`2 - u`1 * x`2) * s1 = 0.R by A1;
     consider s2 being Element of R such that
A5:  s2 in S and
A6:  (y`1 * v`2 - v`1 * y`2) * s2 = 0.R by A2;
A7:  Fracmult(x,y)`1*Fracmult(u,v)`2 -(u`1*x`2)*(y`1*v`2)
       = (x`1*y`1*u`2*v`2) -(u`1*x`2)*(y`1*v`2) by GROUP_1:def 3
      .= (x`1*u`2*y`1*v`2) -(u`1*x`2)*(y`1*v`2) by GROUP_1:def 3
      .= (x`1*u`2)*(y`1*v`2) -(u`1*x`2)*(y`1*v`2) by GROUP_1:def 3
      .= (x`1*u`2 - u`1*x`2)*(y`1*v`2) by VECTSP_1:13;
A8:  (u`1*x`2)*(y`1*v`2)-Fracmult(u,v)`1*Fracmult(x,y)`2
       = (y`1*v`2)*(u`1*x`2)-(v`1*u`1*y`2*x`2) by GROUP_1:def 3
      .= (y`1*v`2)*(u`1*x`2)-(v`1*y`2*u`1*x`2) by GROUP_1:def 3
      .= (y`1*v`2)*(u`1*x`2)-(v`1*y`2)*(u`1*x`2) by GROUP_1:def 3
      .= (y`1*v`2 - v`1*y`2)*(u`1*x`2) by VECTSP_1:13;
A9:   Fracmult(x,y)`1*Fracmult(u,v)`2 - Fracmult(u,v)`1*Fracmult(x,y)`2
       = Fracmult(x,y)`1*Fracmult(u,v)`2 - Fracmult(u,v)`1*Fracmult(x,y)`2
         + 0.R
      .= Fracmult(x,y)`1*Fracmult(u,v)`2 -Fracmult(u,v)`1*Fracmult(x,y)`2
         +(-(u`1*x`2)*(y`1*v`2)+(u`1*x`2)*(y`1*v`2)) by RLVECT_1:5
      .= (Fracmult(x,y)`1*Fracmult(u,v)`2+(-(u`1*x`2)*(y`1*v`2)
          +(u`1*x`2)*(y`1*v`2))) +(-Fracmult(u,v)`1*Fracmult(x,y)`2)
         by RLVECT_1:def 3
      .= (Fracmult(x,y)`1*Fracmult(u,v)`2+(-(u`1*x`2)*(y`1*v`2)))
         +(u`1*x`2)*(y`1*v`2) +(-Fracmult(u,v)`1*Fracmult(x,y)`2)
         by RLVECT_1:def 3
      .= ((x`1*u`2-u`1*x`2)*(y`1*v`2))+((y`1*v`2 - v`1*y`2)*(u`1*x`2))
         by A8,A7,RLVECT_1:def 3;
      reconsider s = s1*s2 as Element of S by A3,A5,C0SP1:def 4;
      reconsider t = x`1*u`2-u`1*x`2 as Element of R;
      reconsider t2 = s2*y`1*v`2 as Element of R;
      (Fracmult(x,y)`1*Fracmult(u,v)`2 - Fracmult(u,v)`1*Fracmult(x,y)`2)*s
       = t*(y`1*v`2)*s+(y`1*v`2-v`1*y`2)*(u`1*x`2)*s by A9,VECTSP_1:def 3
      .= (t*s)*(y`1*v`2)+(y`1*v`2-v`1*y`2)*(u`1*x`2)*s by GROUP_1:def 3
      .= ((0.R)*s2)*(y`1*v`2)+(y`1*v`2-v`1*y`2)*(u`1*x`2)*s by A4,GROUP_1:def 3
      .= ((y`1*v`2-v`1*y`2)*s)*(u`1*x`2) by GROUP_1:def 3
      .= ((0.R)*s1)*(u`1*x`2) by A6, GROUP_1:def 3
      .= 0.R;
      hence thesis;
    end;
