
theorem Th13:
  for I being non degenerated domRing-like commutative Ring for u,
  v,w being Element of Quot.I holds qmult(u,qmult(v,w)) = qmult(qmult(u,v),w) &
  qmult(u,v) = qmult(v,u)
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v,w be Element of Quot.I;
  consider x being Element of Q.I such that
A1: u = QClass.x by Def5;
  consider z being Element of Q.I such that
A2: w = QClass.z by Def5;
  consider y being Element of Q.I such that
A3: v = QClass.y by Def5;
A4: qmult(u,v) = QClass.(pmult(x,y)) by A1,A3,Th10
    .= qmult(v,u) by A1,A3,Th10;
  qmult(u,qmult(v,w)) = qmult(QClass.x,QClass.(pmult(y,z))) by A1,A3,A2,Th10
    .= QClass.(pmult(x,pmult(y,z))) by Th10
    .= QClass.(pmult(pmult(x,y),z)) by Th4
    .= qmult(QClass.(pmult(x,y)),QClass.z) by Th10
    .= qmult(qmult(u,v),w) by A1,A3,A2,Th10;
  hence thesis by A4;
end;
