
theorem Th3:
for L being add-associative right_zeroed right_complementable
            associative distributive non empty doubleLoopStr
for k being even 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 even Element of NAT;
let x be Element of L;
defpred P[Nat] means
   $1 is even implies (power L).(-x,$1) = (power L).(x,$1);
A1: now let k be Nat;
   assume A2: for n being Nat st n < k holds P[n];
   now assume A3: k is even;
     now per cases by NAT_1:23;
     case A4: k = 0;
        hence (power L).(-x,k) = 1_ L by GROUP_1:def 7
                              .= (power L).(x,k) by A4,GROUP_1:def 7;
        end;
     case k = 1;
       hence (power L).(-x,k) = (power L).(x,k) by Lm1,A3;
       end;
     case A5: k >= 2;
       then reconsider k2 = k - 2 as Element of NAT by NAT_1:21;
       k - 1 >= 2 - 1 by A5,XREAL_1:9;
       then reconsider k1 = k - 1 as Element of NAT by INT_1:3;
       A6: k1 + 1 = k;
       A7: k - 2 < k - 0 by XREAL_1:10;
       consider l being Nat such that A8: k = 2*l by A3,ABIAN:def 2;
       A9: 2 * (l - 1) = k - 2 by A8;
       reconsider a = (power L).(-x,k1) as Element of L;
       reconsider b = (power L).(x,k1) as Element of L;
       reconsider y = (power L).(-x,k2) as Element of L;
       reconsider z = (power L).(x,k2) as Element of L;
       A10: (power L).(-x,k2+1) = y * (-x) by GROUP_1:def 7;
       A11: (power L).(x,k2+1) = z * (x) by GROUP_1:def 7;
       thus (power L).(-x,k) = (y * (-x)) * (-x) by A10,A6,GROUP_1:def 7
                            .= (z * (-x)) * (-x) by A7,A2,A9
                            .= z * ((-x) * (-x)) by GROUP_1:def 3
                            .= z * (x * x) by VECTSP_1:10
                            .= b * x by A11,GROUP_1:def 3
                            .= (power L).(x,k) by A6,GROUP_1:def 7;
       end;
     end;
     hence (power L).(-x,k) = (power L).(x,k);
     end;
   hence P[k];
   end;
for k being Nat holds P[k] from NAT_1:sch 4(A1);
hence thesis;
end;
