
theorem Th2:
  for L being add-associative right_zeroed right_complementable
  associative commutative well-unital almost_left_invertible distributive non
degenerated non empty doubleLoopStr for k being Element of NAT holds power(L)
  .(-1_L,k) <> 0.L
proof
  let L be add-associative right_zeroed right_complementable associative
  commutative well-unital almost_left_invertible distributive non degenerated
  non empty doubleLoopStr, k be Element of NAT;
  defpred P[Nat] means power(L).(-1_L,$1) <> 0.L;
A1: now
A2: now
      assume
A3:   -1_L = 0.L;
      1_L = 1_L * 1_L
        .= (-1_L) * (-1_L) by VECTSP_1:10
        .= 0.L by A3;
      hence contradiction;
    end;
    let k be Nat;
     reconsider kk=k as Element of NAT by ORDINAL1:def 12;
A4: power(L).(-1_L,kk+1) = power(L).(-1_L,kk) * (-1_L) by GROUP_1:def 7;
    assume P[k];
    hence P[k+1] by A4,A2,VECTSP_1:12;
  end;
A5: P[0] by GROUP_1:def 7;
  for k be Nat holds P[k] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
