
theorem Th4:
for L being add-associative right_zeroed right_complementable
            associative distributive non empty doubleLoopStr
for k being odd Element of NAT
for x being Element of L
holds (power L).(-x,k) = - ((power L).(x,k))
proof
let L be add-associative right_zeroed right_complementable
         associative distributive non empty doubleLoopStr;
let k be odd Element of NAT;
let x be Element of L;
per cases by NAT_1:14;
suppose k = 0;
  hence (power L).(-x,k) = - (power L).(x,k);
  end;
suppose k >= 1;
  then reconsider k1 = k - 1 as Element of NAT by INT_1:5;
  A1: k1 + 1 = k;
  reconsider a = (power L).(-x,k1) as Element of L;
  reconsider b = (power L).(x,k1) as Element of L;
  (power L).(-x,k1) = (power L).(x,k1) by Th3;
  hence (power L).(-x,k) = b * (-x) by A1,GROUP_1:def 7
                        .= - (b * x) by VECTSP_1:8
                        .= - (power L).(x,k) by A1,GROUP_1:def 7;

  end;
end;
