
theorem Th14:
  for I being non degenerated domRing-like commutative Ring for u
  being Element of Quot.I holds qmult(u,q1.I) = u & qmult(q1.I,u) = u
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: q1.I = QClass.x by Def5;
  consider y being Element of Q.I such that
A2: u = QClass.y by Def5;
A3: x`2 <> 0.I by Th2;
  y`2 <> 0.I by Th2;
  then x`2 * y`2 <> 0.I by A3,VECTSP_2:def 1;
  then reconsider t = [x`1 * y`1, x`2 * y`2] as Element of Q.I by Def1;
  x in q1.I by A1,Th5;
  then
A4: x`1 = x`2 by Def9;
A5: for z being Element of Q.I holds z in QClass.y implies z in QClass.t
  proof
    let z be Element of Q.I;
    assume z in QClass.y;
    then
A6: z`1 * y`2 = z`2 * y`1 by Def4;
    z`1 * t`2 = z`1 * (x`2 * y`2)
      .= (z`2 * y`1) * x`2 by A6,GROUP_1:def 3
      .= z`2 * (x`1 * y`1) by A4,GROUP_1:def 3
      .= z`2 * t`1;
    hence thesis by Def4;
  end;
A7: for z being Element of Q.I holds z in QClass.t implies z in QClass.y
  proof
    let z be Element of Q.I;
    x`2 divides x`2;
    then
A8: x`2 divides (z`1 * y`2) * x`2 by GCD_1:7;
    x`2 divides x`2;
    then
A9: x`2 divides (z`2 * y`1) * x`2 by GCD_1:7;
    assume z in QClass.t;
    then z`1 * t`2 = z`2 * t`1 by Def4;
    then
A10: z`1 * (x`2 * y`2) = z`2 * t`1;
A11: (z`1 * y`2) * x`2 = z`1 * (x`2 * y`2) by GROUP_1:def 3
      .= z`2 * (x`2 * y`1) by A4,A10
      .= (z`2 * y`1) * x`2 by GROUP_1:def 3;
    z`1 * y`2 = (z`1 * y`2) * 1_I
      .= (z`1 * y`2) * (x`2/x`2) by A3,GCD_1:9
      .= ((z`2 * y`1) * x`2) / x`2 by A3,A11,A8,GCD_1:11
      .= (z`2 * y`1) * (x`2/x`2) by A3,A9,GCD_1:11
      .= (z`2 * y`1) * 1_I by A3,GCD_1:9
      .= z`2 * y`1;
    hence thesis by Def4;
  end;
  qmult(u,q1.I) = QClass.(pmult(y,x)) & qmult(q1.I,u) = QClass.(pmult(x,y
  )) by A1,A2,Th10;
  hence thesis by A2,A5,A7,SUBSET_1:3;
end;
