
theorem Th11:
  for L being add-associative right_zeroed right_complementable
distributive non empty doubleLoopStr, m1,m2 being AlgSequence of L holds len(
  m1 * m2) <= min(len m1, len m2)
proof
  let L be add-associative right_zeroed right_complementable distributive non
  empty doubleLoopStr, m1,m2 be AlgSequence of L;
  set p = m1 * m2, k = min(len m1, len m2);
  reconsider k as Element of NAT;
  now
    let i be Nat;
    assume
A1: i >= k;
    per cases by XXREAL_0:15;
    suppose
      k = len m1;
      then m1.i = 0.L by A1,ALGSEQ_1:8;
      hence 0.L = m1.i * m2.i
        .= p.i by LOPBAN_3:def 7;
    end;
    suppose
      k = len m2;
      then m2.i = 0.L by A1,ALGSEQ_1:8;
      hence 0.L = m1.i * m2.i
        .= p.i by LOPBAN_3:def 7;
    end;
  end;
  then k is_at_least_length_of p by ALGSEQ_1:def 2;
  hence thesis by ALGSEQ_1:def 3;
end;
