
theorem Th10:
  for I being non degenerated domRing-like commutative Ring for u,
  v being Element of Q.I holds qmult(QClass.u,QClass.v) = QClass.(pmult(u,v))
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v be Element of Q.I;
  u`2 <> 0.I & v`2 <> 0.I by Th2;
  then u`2 * v`2 <> 0.I by VECTSP_2:def 1;
  then reconsider w = [u`1 * v`1, u`2 * v`2] as Element of Q.I by Def1;
A1: w`1 = u`1 * v`1 & w`2 = u`2 * v`2;
A2: for z being Element of Q.I holds z in qmult(QClass.u,QClass.v) implies z
  in QClass.(pmult(u,v))
  proof
    let z be Element of Q.I;
    assume z in qmult(QClass.u,QClass.v);
    then consider a,b being Element of Q.I such that
A3: a in QClass.u and
A4: b in QClass.v and
A5: z`1 * (a`2 * b`2) = z`2 * (a`1 * b`1) by Def7;
A6: b`1 * v`2 = b`2 * v`1 by A4,Def4;
A7: a`1 * u`2 = a`2 * u`1 by A3,Def4;
    now
      per cases;
      case
A8:     a`1 = 0.I;
        then a`1 * b`1 = 0.I;
        then
A9:     z`2 * (a`1 * b`1) = 0.I;
A10:    a`2 <> 0.I by Th2;
        b`2 <> 0.I by Th2;
        then a`2 * b`2 <> 0.I by A10,VECTSP_2:def 1;
        then
A11:    z`1 = 0.I by A5,A9,VECTSP_2:def 1;
        a`1 * u`2 = 0.I by A8;
        then u`1 = 0.I by A7,A10,VECTSP_2:def 1;
        then z`2 * (u`1 * v`1) = z`2 * 0.I
          .= 0.I
          .= z`1 * (u`2 * v`2) by A11;
        hence thesis by A1,Def4;
      end;
      case
A12:    b`1 = 0.I;
        then a`1 * b`1 = 0.I;
        then
A13:    z`2 * (a`1 * b`1) = 0.I;
A14:    b`2 <> 0.I by Th2;
        a`2 <> 0.I by Th2;
        then a`2 * b`2 <> 0.I by A14,VECTSP_2:def 1;
        then
A15:    z`1 = 0.I by A5,A13,VECTSP_2:def 1;
        b`1 * v`2 = 0.I by A12;
        then v`1 = 0.I by A6,A14,VECTSP_2:def 1;
        then z`2 * (u`1 * v`1) = z`2 * 0.I
          .= 0.I
          .= z`1 * (u`2 * v`2) by A15;
        hence thesis by A1,Def4;
      end;
      case
A16:    a`1 <> 0.I & b`1 <> 0.I;
        a`1 * b`1 divides a`1 * b`1;
        then
A17:    a`1 * b`1 divides ((z`2 * u`1) * v`1) * (a`1 * b`1) by GCD_1:7;
A18:    a`1 * b`1 <> 0.I by A16,VECTSP_2:def 1;
A19:    b`1 divides b`2 * v`1 by A6,GCD_1:def 1;
        then
A20:    v`2 = (b`2 * v`1) / b`1 by A6,A16,GCD_1:def 4;
A21:    a`1 divides a`2 * u`1 by A7,GCD_1:def 1;
        then
A22:    a`1 * b`1 divides (a`2 * u`1) * (b`2 * v`1) by A19,GCD_1:3;
        then
A23:    a`1 * b`1 divides z`1 * ((a`2 * u`1) * (b`2 * v`1)) by GCD_1:7;
        u`2 = (a`2 * u`1) / a`1 by A7,A16,A21,GCD_1:def 4;
        then
        z`1 * (u`2 * v`2) = z`1 * (((a`2 * u`1) * (b`2 * v`1)) / (a`1 * b
        `1)) by A16,A21,A19,A20,GCD_1:14
          .= (z`1 * ((a`2 * u`1) * (b`2 * v`1))) / (a`1 * b`1) by A18,A22,A23,
GCD_1:11
          .= (z`1 * (((u`1 * a`2) * b`2) * v`1)) / (a`1 * b`1) by GROUP_1:def 3
          .= (z`1 * (((a`2 * b`2) * u`1) * v`1)) / (a`1 * b`1) by GROUP_1:def 3
          .= ((z`1 * ((a`2 * b`2) * u`1)) * v`1) / (a`1 * b`1) by GROUP_1:def 3
          .= (((z`2 * (a`1 * b`1)) * u`1) * v`1) / (a`1 * b`1) by A5,
GROUP_1:def 3
          .= (((z`2 * u`1) * (a`1 * b`1)) * v`1) / (a`1 * b`1) by GROUP_1:def 3
          .= (((z`2 * u`1) * v`1) * (a`1 * b`1)) / (a`1 * b`1) by GROUP_1:def 3
          .= ((z`2 * u`1) * v`1) * ((a`1 * b`1) / (a`1 * b`1)) by A18,A17,
GCD_1:11
          .= ((z`2 * u`1) * v`1) * 1_I by A18,GCD_1:9
          .= (z`2 * u`1) * v`1
          .= z`2 * (u`1 * v`1) by GROUP_1:def 3;
        hence thesis by A1,Def4;
      end;
    end;
    hence thesis;
  end;
  for z being Element of Q.I holds z in QClass.(pmult(u,v)) implies z in
  qmult(QClass.u,QClass.v)
  proof
    let z be Element of Q.I;
    assume z in QClass.(pmult(u,v));
    then
A24: z`1 * (u`2 * v`2) = z`2 * (u`1 * v`1) by A1,Def4;
    u in QClass.u & v in QClass.v by Th5;
    hence thesis by A24,Def7;
  end;
  hence thesis by A2,SUBSET_1:3;
end;
