
theorem Th4:
  for L be non degenerated non empty multLoopStr_0 holds len 1_.( L) = 1
proof
  let L be non degenerated non empty multLoopStr_0;
A1: now
    let i be Nat;
    assume that
A2: i is_at_least_length_of 1_.(L) and
A3: 0+1 > i;
    0 >= i by A3,NAT_1:13;
    then (1_.(L)).0 = 0.L by A2;
    hence contradiction by POLYNOM3:30;
  end;
  for i be Nat st i >= 1 holds (1_.(L)).i = 0.L by POLYNOM3:30;
  then 1 is_at_least_length_of 1_.(L);
  hence thesis by A1,ALGSEQ_1:def 3;
end;
