
theorem Th4:
  for L being add-associative right_zeroed right_complementable
  well-unital distributive non empty doubleLoopStr for k being Element of NAT
  holds power(L).(-1_L,2*k) = 1_L & power(L).(-1_L,2*k+1) = -1_L
proof
  let L be add-associative right_zeroed right_complementable well-unital
  distributive non empty doubleLoopStr, k be Element of NAT;
  defpred P[Nat] means power(L).(-1_L,2*$1) = 1_L & power(L).(-1_L,2*$1+1) = -
  1_L;
A1: now
    let k be Nat;
    assume
A2: P[k];
A3: power(L).(-1_L,2*(k+1)) = power(L).(-1_L,(2*k+1)+1)
      .= power(L).(-1_L,2*k+1) * (-1_L) by GROUP_1:def 7
      .= - (1_L * (-1_L)) by A2,VECTSP_1:9
      .= - (- 1_L)
      .= 1_L by RLVECT_1:17;
    power(L).(-1_L,2*(k+1)+1) = power(L).(-1_L,2*(k+1)) * (-1_L) by
GROUP_1:def 7
      .= - (1_L) by A3;
    hence P[k+1] by A3;
  end;
  power(L).(-1_L,2*0+1) = power(L).(-1_L,0) * (-1_L) by GROUP_1:def 7
    .= 1_L * (-1_L) by GROUP_1:def 7
    .= -1_L;
  then
A4: P[0] by GROUP_1:def 7;
  for k be Nat holds P[k] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
