
theorem Th17:
  for L be unital non empty doubleLoopStr for x be Element of L
  holds eval(0_.(L),x) = 0.L
proof
  let L be unital non empty doubleLoopStr;
  let x be Element of L;
  consider F be FinSequence of the carrier of L such that
A1: eval(0_.(L),x) = Sum F and
A2: len F = len 0_.(L) and
  for n be Element of NAT st n in dom F holds F.n = (0_.(L)).(n-'1) * (
  power L).(x,n-'1) by Def2;
  len F = 0 by A2,Th3;
  then F = <*>the carrier of L;
  hence thesis by A1,RLVECT_1:43;
end;
