
theorem Th35:
  for L being add-associative right_zeroed right_complementable
  distributive well-unital commutative associative non degenerated non empty
  doubleLoopStr, r being Element of L, q being non-zero Polynomial of L holds
  len (<%r, 1.L%>*'q) = len q + 1
proof
  let L be add-associative right_zeroed right_complementable distributive
well-unital commutative associative non degenerated non empty doubleLoopStr,
  r be Element of L, q being non-zero Polynomial of L;
  set p = <%r, 1.L%>;
A1: p.(len p -'1) * q.(len q -'1) = p.(1+1-'1) * q.(len q -'1) by POLYNOM5:40
    .= p.(1) * q.(len q -'1) by NAT_D:34
    .= 1.L * q.(len q -'1) by POLYNOM5:38
    .= q.(len q -'1);
  len <%r, 1.L%> = 2 & len q > 0 by Th14,POLYNOM5:40;
  hence len (<%r, 1.L%>*'q) = len q +(1+1)-1 by A1,Th15,POLYNOM4:10
    .= len q +1;
end;
