
theorem Th17:
  for I being non degenerated domRing-like commutative Ring for u
being Element of Quot.I holds qadd(u,qaddinv(u)) = q0.I & qadd(qaddinv(u),u) =
  q0.I
proof
  let I be non degenerated domRing-like commutative Ring;
  let u be Element of Quot.I;
  consider x being Element of Q.I such that
A1: qaddinv(u) = QClass.x by Def5;
  x in qaddinv(u) by A1,Th5;
  then consider a being Element of Q.I such that
A2: a in u and
A3: x`1 * a`2 = x`2 * (-(a`1)) by Def10;
  consider y being Element of Q.I such that
A4: u = QClass.y by Def5;
  x`2 <> 0.I & y`2 <> 0.I by Th2;
  then x`2 * y`2 <> 0.I by VECTSP_2:def 1;
  then reconsider t = [y`1 * x`2 + x`1 * y`2, x`2 * y`2] as Element of Q.I by
Def1;
A5: a`2 <> 0.I by Th2;
  y in u by A4,Th5;
  then
A6: y`1 * a`2 = a`1 * y`2 by A2,Th7;
  t`1 * a`2 = (y`1 * x`2 + x`1 * y`2) * a`2
    .= (y`1 * x`2) * a`2 + (x`1 * y`2) * a`2 by VECTSP_1:def 3
    .= x`2 * (a`1 * y`2) + (x`1 * y`2) * a`2 by A6,GROUP_1:def 3
    .= x`2 * (a`1 * y`2) + (x`2 * (-(a`1))) * y`2 by A3,GROUP_1:def 3
    .= x`2 * (a`1 * y`2) + (-(x`2 * a`1)) * y`2 by GCD_1:48
    .= x`2 * (a`1 * y`2) + (-(x`2 * a`1) * y`2) by GCD_1:48
    .= (x`2 * a`1) * y`2 + (-(x`2 * a`1 * y`2)) by GROUP_1:def 3
    .= 0.I by RLVECT_1:def 10;
  then
A7: t`1 = 0.I by A5,VECTSP_2:def 1;
A8: for z being Element of Q.I holds z in q0.I implies z in QClass.t
  proof
    let z be Element of Q.I;
    assume z in q0.I;
    then z`1 = 0.I by Def8;
    then
A9: z`1 * t`2 = 0.I;
    z`2 * t`1 = 0.I by A7;
    hence thesis by A9,Def4;
  end;
A10: t`2 <> 0.I by Th2;
A11: for z being Element of Q.I holds z in QClass.t implies z in q0.I
  proof
    let z be Element of Q.I;
    assume z in QClass.t;
    then
A12: z`1 * t`2 = z`2 * t`1 by Def4;
    z`2 * t`1 = 0.I by A7;
    then z`1 = 0.I by A10,A12,VECTSP_2:def 1;
    hence thesis by Def8;
  end;
  qadd(u,qaddinv(u)) = QClass.(padd(y,x)) & qadd(qaddinv(u),u) = QClass.(
  padd( x,y)) by A1,A4,Th9;
  hence thesis by A11,A8,SUBSET_1:3;
end;
