
theorem Th7:
  for I being non degenerated domRing-like commutative Ring for u,v
  being Element of Q.I holds (ex w being Element of Quot.I st u in w & v in w)
  implies u`1 * v`2 = v`1 * u`2
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v be Element of Q.I;
  given w being Element of Quot.I such that
A1: u in w and
A2: v in w;
  consider z being Element of Q.I such that
A3: w = QClass.z by Def5;
A4: u`1 * z`2 = z`1 * u`2 by A1,A3,Def4;
  z`2 divides z`2;
  then
A5: z`2 divides (v`2 * u`1) * z`2 by GCD_1:7;
A6: v`1 * z`2 = z`1 * v`2 by A2,A3,Def4;
  then
A7: z`2 divides z`1 * v`2 by GCD_1:def 1;
  then
A8: z`2 divides (z`1 * v`2) * u`2 by GCD_1:7;
A9: z`2 <> 0.I by Th2;
  hence v`1 * u`2 = ((z`1 * v`2)/z`2) * u`2 by A6,A7,GCD_1:def 4
    .= ((z`1 * v`2) * u`2) / z`2 by A7,A8,A9,GCD_1:11
    .= (v`2 * (u`1 * z`2)) / z`2 by A4,GROUP_1:def 3
    .= ((v`2 * u`1) * z`2) / z`2 by GROUP_1:def 3
    .= (v`2 * u`1) * (z`2/z`2) by A5,A9,GCD_1:11
    .= (u`1 * v`2) * 1_I by A9,GCD_1:9
    .= u`1 * v`2;
end;
