
theorem Th11:
  for I being non degenerated domRing-like commutative Ring for u,
v,w being Element of Quot.I holds qadd(u,qadd(v,w)) = qadd(qadd(u,v),w) & qadd(
  u,v) = qadd(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: qadd(u,v) = QClass.(padd(x,y)) by A1,A3,Th9
    .= qadd(v,u) by A1,A3,Th9;
  qadd(u,qadd(v,w)) = qadd(QClass.x,QClass.(padd(y,z))) by A1,A3,A2,Th9
    .= QClass.(padd(x,padd(y,z))) by Th9
    .= QClass.(padd(padd(x,y),z)) by Th3
    .= qadd(QClass.(padd(x,y)),QClass.z) by Th9
    .= qadd(qadd(u,v),w) by A1,A3,A2,Th9;
  hence thesis by A4;
end;
