 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th27:
  for p,q being Element of the carrier of Polynom-Ring INT.Ring
   holds ^(p + q) = ^p + ^q & ^(p * q) = (^p) * (^q)
   proof
     let p,q be Element of the carrier of Polynom-Ring INT.Ring;
A3:  ^(p + q) = ~(p + q)
     .= ~p + ~q by POLYNOM3:def 10
     .= ~^p + ~^q by FIELD_4:15 .= ~(^p + ^q) by POLYNOM3:def 10
     .= ^p + (^q);
     ^(p * q) = ~(p * q)
     .= ~p *' ~q by POLYNOM3:def 10
     .= (~^p) *' (~^q) by FIELD_4:17 .= ~(^p * (^q)) by POLYNOM3:def 10
     .= (^p) * (^q);
     hence thesis by A3;
   end;
