
theorem Th47:
  for I being non degenerated domRing-like commutative Ring for x
being Element of the_Field_of_Quotients(I) st x <> 0.the_Field_of_Quotients(I)
for a being Element of I st a <> 0.I for u being Element of Q.I st x = QClass.u
  & u = [a,1.I] for v being Element of Q.I st v = [1.I,a] holds x" = QClass.v
proof
  let I be non degenerated domRing-like commutative Ring;
  let x be Element of the_Field_of_Quotients(I);
  assume
A1: x <> 0.the_Field_of_Quotients(I);
  let a be Element of I;
  assume
A2: a <> 0.I;
  then reconsider res = [a,a] as Element of Q.I by Def1;
A3: for u being object holds u in QClass.res implies u in q1.I
  proof
    let u be object;
    assume
A4: u in QClass.res;
    then reconsider u as Element of Q.I;
    u`1 * a = u`1 * res`2
      .= u`2 * res`1 by A4,Def4
      .= u`2 * a;
    then u`1 = u`2 by A2,GCD_1:1;
    hence thesis by Def9;
  end;
  for u being object holds u in q1.I implies u in QClass.res
  proof
    let u be object;
    assume
A5: u in q1.I;
    then reconsider u as Element of Q.I;
    u`1 * res`2 = u`1 * a
      .= u`2 * a by A5,Def9
      .= u`2 * res`1;
    hence thesis by Def4;
  end;
  then
A6: q1.I = QClass.res by A3,TARSKI:2;
  let u be Element of Q.I;
  assume that
A7: x = QClass.u and
A8: u = [a,1.I];
  let v be Element of Q.I;
  assume
A9: v = [1.I,a];
  pmult(u,v) = [a * v`1, u`2 * v`2] by A8
    .= [a * 1.I, u`2 * v`2] by A9
    .= [a * 1.I, 1.I * v`2] by A8
    .= [a * 1.I, 1.I * a] by A9
    .= [a, 1.I * a]
    .= [a, a];
  then
A10: qmult(QClass.u,QClass.v) = 1.the_Field_of_Quotients(I) by A6,Th10;
  reconsider y = QClass.v as Element of the_Field_of_Quotients(I);
  reconsider y as Element of the_Field_of_Quotients(I);
  qmult(QClass.u,QClass.v) = x * y by A7,Def13;
  hence thesis by A1,A10,VECTSP_1:def 10;
end;
