
theorem Th9:
  for I being non degenerated domRing-like commutative Ring for u,
  v being Element of Q.I holds qadd(QClass.u,QClass.v) = QClass.(padd(u,v))
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v be Element of Q.I;
  u`2 <> 0.I & v`2 <> 0.I by Th2;
  then u`2 * v`2 <> 0.I by VECTSP_2:def 1;
  then reconsider w = [u`1 * v`2 + v`1 * u`2, u`2 * v`2] as Element of Q.I by
Def1;
A1: w`1 = u`1 * v`2 + v`1 * u`2 & w`2 = u`2 * v`2;
A2: for z being Element of Q.I holds z in qadd(QClass.u,QClass.v) implies z
  in QClass.(padd(u,v))
  proof
    let z be Element of Q.I;
    assume z in qadd(QClass.u,QClass.v);
    then consider a,b being Element of Q.I such that
A3: a in QClass.u and
A4: b in QClass.v and
A5: z`1 * (a`2 * b`2) = z`2 * (a`1 * b`2 + b`1 * a`2) by Def6;
A6: a`1 * u`2 = a`2 * u`1 by A3,Def4;
A7: b`1 * v`2 = b`2 * v`1 by A4,Def4;
    a`2 * b`2 divides a`2 * b`2;
    then
A8: a`2 * b`2 divides (z`2 * ((u`1 * v`2)+(v`1 * u`2))) * (a`2 * b`2) by
GCD_1:7;
A9: a`2 * b`2 divides z`2 * (a`1 * b`2 + b`1 * a`2) by A5,GCD_1:def 1;
    then
A10: a`2 * b`2 divides (z`2 * (a`1 * b`2 + b`1 * a`2)) * (u`2 * v`2) by GCD_1:7
;
    a`2 <> 0.I & b`2 <> 0.I by Th2;
    then
A11: a`2 * b`2 <> 0.I by VECTSP_2:def 1;
    then z`1 = (z`2 * (a`1 * b`2 + b`1 * a`2)) / (a`2 * b`2) by A5,A9,
GCD_1:def 4;
    then
    z`1 * (u`2 * v`2) = ((z`2 * (a`1 * b`2 + b`1 * a`2)) * (u`2 * v`2)) /
    (a`2 * b`2) by A11,A9,A10,GCD_1:11
      .= (z`2 * ((a`1 * b`2 + b`1 * a`2) * (u`2 * v`2))) / (a`2 * b`2) by
GROUP_1:def 3
      .= (z`2 * ((a`1 * b`2) * (u`2 * v`2) + (b`1 * a`2) * (u`2 * v`2))) / (
    a`2 * b`2) by VECTSP_1:def 3
      .= (z`2 * (b`2 * (a`1 * (u`2 * v`2)) + (b`1 * a`2) * (u`2 * v`2))) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * (b`2 * ((a`2 * u`1) * v`2) + (b`1 * a`2) * (u`2 * v`2))) / (
    a`2 * b`2) by A6,GROUP_1:def 3
      .= (z`2 * (b`2 * ((a`2 * u`1) * v`2) + a`2 * (b`1 * (v`2 * u`2)))) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * (b`2 * ((a`2 * u`1) * v`2) + a`2 * ((b`2 * v`1) * u`2))) / (
    a`2 * b`2) by A7,GROUP_1:def 3
      .= (z`2 * ((b`2 * (a`2 * u`1)) * v`2 + a`2 * ((b`2 * v`1) * u`2))) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * ((u`1 * (b`2 * a`2)) * v`2 + a`2 * ((b`2 * v`1) * u`2))) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * ((u`1 * v`2) * (b`2 * a`2) + a`2 * ((b`2 * v`1) * u`2))) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * ((u`1 * v`2) * (b`2 * a`2) + (a`2 * (b`2 * v`1)) * u`2)) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * ((u`1 * v`2) * (a`2 * b`2) + (v`1 * (a`2 * b`2)) * u`2)) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * ((u`1 * v`2) * (a`2 * b`2) + (v`1 * u`2) * (a`2 * b`2))) / (
    a`2 * b`2) by GROUP_1:def 3
      .= (z`2 * (((u`1 * v`2) + (v`1 * u`2)) * (a`2 * b`2))) / (a`2 * b`2)
    by VECTSP_1:def 3
      .= ((z`2 * ((u`1 * v`2) + (v`1 * u`2))) * (a`2 * b`2)) / (a`2 * b`2)
    by GROUP_1:def 3
      .= (z`2 * ((u`1 * v`2) + (v`1 * u`2))) * ((a`2 * b`2)/(a`2 * b`2)) by A11
,A8,GCD_1:11
      .= (z`2 * ((u`1 * v`2) + (v`1 * u`2))) * 1_I by A11,GCD_1:9
      .= z`2 * ((u`1 * v`2) + (v`1 * u`2));
    hence thesis by A1,Def4;
  end;
  for z being Element of Q.I holds z in QClass.(padd(u,v)) implies z in
  qadd(QClass.u,QClass.v)
  proof
    let z be Element of Q.I;
    assume z in QClass.(padd(u,v));
    then
A12: z`1 * (u`2 * v`2) = z`2 * (u`1 * v`2 + v`1 * u`2) by A1,Def4;
    u in QClass.u & v in QClass.v by Th5;
    hence thesis by A12,Def6;
  end;
  hence thesis by A2,SUBSET_1:3;
end;
