reserve n, k, r, m, i, j for Nat;

theorem Th7:
  ((-1) to_power (-n)) ^2 = 1
proof
  ((-1) to_power (-n)) ^2 = ((-1) #Z (-n)) ^2 by POWER:def 2
    .= (1 / (-1) #Z n) ^2 by PREPOWER:41
    .= (1 / ((-1) #Z n)) to_power 2 by POWER:46
    .= (1 / ((-1) #Z n)) |^2 by POWER:41
    .= 1 / (((-1) #Z n) |^ 2) by PREPOWER:7
    .= 1 / (((-1) #Z n) #Z 2) by PREPOWER:36
    .= 1 / ((-1) #Z (n*2)) by PREPOWER:45
    .= 1 / ((-1) |^ (2*n)) by PREPOWER:36
    .= 1 / (1 |^ (2*n)) by WSIERP_1:2
    .= 1 / ((1 |^2) |^ n)
    .= 1 / (1 |^ n)
    .= 1 / 1;
  hence thesis;
end;
