
theorem Th1:
  for n being Nat, L being well-unital domRing-like non
degenerated non empty doubleLoopStr, x being Element of L st x <> 0.L holds x
  |^ n <> 0.L
proof
  let n be Nat;
  let L be well-unital domRing-like non degenerated non empty doubleLoopStr,
  x being Element of L;
  defpred P[Nat] means x |^ $1 <> 0.L;
  assume
A1: x <> 0.L;
A2: now
    let n be Nat;
    assume P[n];
    then (x |^ n) * x <> 0.L by A1,VECTSP_2:def 1;
    hence P[n+1] by GROUP_1:def 7;
  end;
  x |^ 0 = 1_L by BINOM:8;
  then
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
