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 Th29:
  (x+y)*z, x*z + y*z Fr_Eq S
   proof
A1:  (x`1*z`1)*(y`2*z`2) = x`1*z`1*y`2*z`2 by GROUP_1:def 3
     .= ((x`1*y`2)*z`1)*z`2 by GROUP_1:def 3
     .= (x`1*y`2)*(z`1*z`2) by GROUP_1:def 3;
     (y`1*z`1)*(x`2*z`2) = y`1*z`1*x`2*z`2 by GROUP_1:def 3
     .= ((y`1*x`2)*z`1)*z`2 by GROUP_1:def 3
     .= (y`1*x`2)*(z`1*z`2) by GROUP_1:def 3; then
A3:  Fracadd(Fracmult(x,z),Fracmult(y,z))
     = [(x`1*y`2 + y`1*x`2)*(z`1*z`2),(x`2*z`2)*(y`2*z`2)]
     by A1,VECTSP_1:def 7;
     Fracmult(Fracadd(x,y),z)`1 * Fracadd(Fracmult(x,z),Fracmult(y,z))`2
     = ((x`1*y`2 + y`1*x`2)*z`1)*(z`2*x`2*y`2*z`2) by GROUP_1:def 3
     .= ((x`1*y`2 + y`1*x`2)*z`1)*(z`2*(x`2*y`2)*z`2) by GROUP_1:def 3
     .= (x`1*y`2 + y`1*x`2)*z`1*z`2*((x`2*y`2)*z`2) by GROUP_1:def 3
     .= Fracadd(Fracmult(x,z),Fracmult(y,z))`1* Fracmult(Fracadd(x,y),z)`2
     by A3,GROUP_1:def 3; then
A5:  (Fracmult(Fracadd(x,y),z)`1 * Fracadd(Fracmult(x,z),Fracmult(y,z))`2  -
     Fracadd(Fracmult(x,z),Fracmult(y,z))`1* Fracmult(Fracadd(x,y),z)`2)*1.R
     = 0.R by RLVECT_1:5;
     1.R is Element of S by C0SP1:def 4;
     hence thesis by A5;
   end;
