
theorem Th9:
  for L being add-associative right_zeroed right_complementable
  distributive commutative associative well-unital domRing-like non empty
doubleLoopStr, p,q being Polynomial of L st len p > 0 & len q > 0 holds len(p
  *'q) <= len p + len q
proof
  let L be add-associative right_zeroed right_complementable distributive
  commutative associative well-unital domRing-like non empty doubleLoopStr;
  let p,q be Polynomial of L;
  assume that
A1: len p > 0 and
A2: len q > 0;
A3: (len p + len q) - 1 <= (len p + len q) - 0 by XREAL_1:13;
  len q + 1 > 0 + 1 by A2,XREAL_1:6;
  then len q >= 1 by NAT_1:13;
  then
A4: len q - 1 >= 1 - 1 by XREAL_1:13;
  q.(len(q)-1) <> 0.L by A2,Th8;
  then
A5: q.(len(q)-'1) <> 0.L by A4,XREAL_0:def 2;
  len p + 1 > 0 + 1 by A1,XREAL_1:6;
  then len p >= 1 by NAT_1:13;
  then
A6: len p - 1 >= 1 - 1 by XREAL_1:13;
  p.(len(p)-1) <> 0.L by A1,Th8;
  then p.(len(p)-'1) <> 0.L by A6,XREAL_0:def 2;
  then p.(len p -'1) * q.(len q -'1) <> 0.L by A5,VECTSP_2:def 1;
  hence thesis by A3,POLYNOM4:10;
end;
