
theorem
  for L being add-associative right_zeroed right_complementable
distributive commutative associative well-unital non empty doubleLoopStr, p,
q being Polynomial of L st p.(len p -'1) * q.(len q -'1) <> 0.L holds 0 < len (
  p*'q)
proof
  let L be add-associative right_zeroed right_complementable distributive
  commutative associative well-unital non empty doubleLoopStr, p, q being
  Polynomial of L;
  assume
A1: p.(len p -'1) * q.(len q -'1) <> 0.L;
  now
    assume len q <= 0;
    then len q = 0;
    then q = 0_. L by POLYNOM4:5;
    then q.(len q -'1) = 0.L by FUNCOP_1:7;
    hence contradiction by A1;
  end;
  then
A2: 0 qua Nat+1 <= len q by NAT_1:13;
  now
    assume len p <= 0;
    then len p = 0;
    then p = 0_. L by POLYNOM4:5;
    then p.(len p -'1) = 0.L by FUNCOP_1:7;
    hence contradiction by A1;
  end;
  then 0 qua Nat+1 <= len p by NAT_1:13;
  then len p + len q >= 1+1 by A2,XREAL_1:7;
  then len p + len q -1 >= 1+1-1 by XREAL_1:9;
  hence thesis by A1,POLYNOM4:10;
end;
