
theorem
  for L being add-associative right_zeroed right_complementable
distributive domRing-like non empty doubleLoopStr, m1,m2 being AlgSequence of
  L st len m1 = len m2 holds len(m1 * m2) = len m1
proof
  let L be add-associative right_zeroed right_complementable distributive
  domRing-like non empty doubleLoopStr, m1,m2 be AlgSequence of L;
  set p = m1 * m2;
  assume
A1: len m1 = len m2;
A2: now
    per cases;
    case
      len m1 = 0;
      hence len p >= len m1;
    end;
    case
      len m1 <> 0;
      then len m1 >= 0 + 1 by NAT_1:13;
      then (len m1) - 1 >= 1 - 1 by XREAL_1:9;
      then reconsider l = (len m1) - 1 as Element of NAT by INT_1:3;
A3:   l + 1 = len m1 + 0;
      then m1.l <> 0.L & m2.l <> 0.L by A1,ALGSEQ_1:10;
      then m1.l * m2.l <> 0.L by VECTSP_2:def 1;
      then p.l <> 0.L by LOPBAN_3:def 7;
      hence len p >= len m1 by A3,Th7;
    end;
  end;
  min(len m1, len m2) = len m1 by A1;
  then len p <= len m1 by Th11;
  hence thesis by A2,XXREAL_0:1;
end;
