
theorem Th30:
  for L being add-associative right_zeroed right_complementable
distributive non empty doubleLoopStr, p, q being Polynomial of L st len p = 0
  or len q = 0 holds len (p*'q) = 0
proof
  let L be add-associative right_zeroed right_complementable distributive non
  empty doubleLoopStr, p, q being Polynomial of L;
  assume
A1: len p = 0 or len q = 0;
  per cases by A1;
  suppose
    len p = 0;
    then p = 0_. L by POLYNOM4:5;
    then p*'q = 0_. L by POLYNOM4:2;
    hence thesis by POLYNOM4:3;
  end;
  suppose
    len q = 0;
    then q = 0_. L by POLYNOM4:5;
    then p*'q = 0_. L by Th17;
    hence thesis by POLYNOM4:3;
  end;
end;
