
theorem Th8:
  for I being non degenerated domRing-like commutative Ring for u,v
  being Element of Quot.I holds u meets v implies u = v
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v be Element of Quot.I;
  consider x being Element of Q.I such that
A1: u = QClass.x by Def5;
  assume u /\ v <> {};
  then u meets v;
  then consider w being object such that
A2: w in u and
A3: w in v by XBOOLE_0:3;
  consider y being Element of Q.I such that
A4: v = QClass.y by Def5;
  reconsider w as Element of Q.I by A2;
A5: w`1 * y`2 = w`2 * y`1 by A4,A3,Def4;
A6: for z being Element of Q.I holds z in QClass.x implies z in QClass.y
  proof
    let z be Element of Q.I;
    w`2 divides w`2;
    then
A7: w`2 divides (z`2 * y`1) * w`2 by GCD_1:7;
    assume z in QClass.x;
    then
A8: z`1 * w`2 = z`2 * w`1 by A1,A2,Th7;
    then
A9: w`2 divides z`2 * w`1 by GCD_1:def 1;
    then
A10: w`2 divides (z`2 * w`1) * y`2 by GCD_1:7;
A11: w`2 <> 0.I by Th2;
    then z`1 * y`2 = ((z`2 * w`1)/w`2) * y`2 by A8,A9,GCD_1:def 4
      .= ((z`2 * w`1) * y`2) / w`2 by A9,A10,A11,GCD_1:11
      .= (z`2 * (w`2 * y`1)) / w`2 by A5,GROUP_1:def 3
      .= ((z`2 * y`1) * w`2) / w`2 by GROUP_1:def 3
      .= (z`2 * y`1) * (w`2/w`2) by A7,A11,GCD_1:11
      .= (z`2 * y`1) * 1_I by A11,GCD_1:9
      .= z`2 * y`1;
    hence thesis by Def4;
  end;
A12: w`1 * x`2 = w`2 * x`1 by A1,A2,Def4;
  for z being Element of Q.I holds z in QClass.y implies z in QClass.x
  proof
    let z be Element of Q.I;
    w`2 divides w`2;
    then
A13: w`2 divides (z`2 * x`1) * w`2 by GCD_1:7;
    assume z in QClass.y;
    then
A14: z`1 * w`2 = z`2 * w`1 by A4,A3,Th7;
    then
A15: w`2 divides z`2 * w`1 by GCD_1:def 1;
    then
A16: w`2 divides (z`2 * w`1) * x`2 by GCD_1:7;
A17: w`2 <> 0.I by Th2;
    then z`1 * x`2 = ((z`2 * w`1)/w`2) * x`2 by A14,A15,GCD_1:def 4
      .= ((z`2 * w`1) * x`2) / w`2 by A15,A16,A17,GCD_1:11
      .= (z`2 * (w`2 * x`1)) / w`2 by A12,GROUP_1:def 3
      .= ((z`2 * x`1) * w`2) / w`2 by GROUP_1:def 3
      .= (z`2 * x`1) * (w`2/w`2) by A13,A17,GCD_1:11
      .= (z`2 * x`1) * 1_I by A17,GCD_1:9
      .= z`2 * x`1;
    hence thesis by Def4;
  end;
  hence thesis by A1,A4,A6,SUBSET_1:3;
end;
