reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem Th2:
  n is odd implies (-a) |^ n = - a |^ n
proof
  assume n is odd;
  then consider m such that
A1: n=2*m+1 by ABIAN:9;
  thus
  (-a) |^ n = (-a) |^ (2*m) * (-a) by A1,NEWTON:6
    .= a |^ (2*m) * (-a) by Th1
    .= -a |^ (2*m) * a
    .= -a |^ n by A1,NEWTON:6;
end;
