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 Th1:
  n is even implies (-a) |^ n = a |^ n
proof
  given m such that
A1: n=2*m;
  thus
  (-a) |^ n = ((-a) |^ (1+1)) |^ m by A1,NEWTON:9
    .= ((-a) |^ (0+1) * (-a) |^ (0+1)) |^ m by NEWTON:8
    .= ((-a) |^ 0 * (-a) * (-a) |^ (0+1)) |^ m by NEWTON:6
    .= ((-a) |^ 0 * (-a) * ((-a) |^ 0*(-a))) |^ m by NEWTON:6
    .= ((-a) GeoSeq.0 * (-a) * ((-a) |^ 0 * (-a))) |^ m by PREPOWER:def 1
    .= ((-a) GeoSeq.0 * (-a) * ((-a) GeoSeq.0 * (-a))) |^ m by PREPOWER:def 1
    .= ((-a) GeoSeq.0 * (-a) * (1 * (-a))) |^ m by PREPOWER:3
    .= (1*(-a) * (1*(-a))) |^ m by PREPOWER:3
    .= (a * a) |^ m
    .= ((a GeoSeq.0*a) * (1*a)) |^ m by PREPOWER:3
    .= ((a GeoSeq.0*a) * (a GeoSeq.0*a)) |^ m by PREPOWER:3
    .= ((a GeoSeq.0*a) * (a |^ 0*a)) |^ m by PREPOWER:def 1
    .= ((a |^ 0*a) * (a |^ 0*a)) |^ m by PREPOWER:def 1
    .= ((a |^ 0*a) * a |^ (0+1)) |^ m by NEWTON:6
    .= (a |^ (0+1) * a |^ (0+1)) |^ m by NEWTON:6
    .= (a |^ (1+1)) |^ m by NEWTON:8
    .= a |^ n by A1,NEWTON:9;
end;
