
theorem Th33:
  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 1 < len p & 1 < len q holds len p
  < len (p*'q)
proof
  let L be add-associative right_zeroed right_complementable distributive
commutative associative well-unital domRing-like non empty doubleLoopStr, p,
  q be Polynomial of L such that
A1: 1 < len p and
A2: 1 < len q;
  p.(len p -'1) <> 0.L & q.(len q -'1)<>0.L by A1,A2,Th15;
  then p.(len p -'1) * q.(len q -'1)<>0.L by VECTSP_2:def 1;
  then
A3: len (p*'q) = len p + len q - 1 by POLYNOM4:10;
  len q - 1 > 1-1 by A2,XREAL_1:14;
  then len p + (len q - 1) > 0 qua Nat+len p by XREAL_1:8;
  hence thesis by A3;
end;
