reserve n for Nat,
  a,b for Real,
  s for Real_Sequence;

theorem Th1:
  |.(-1)|^n.| = 1
proof
  per cases;
  suppose
A1: n is even;
    (-1)|^n =(-1) to_power n by POWER:41
      .=1 to_power n by A1,POWER:47
      .=1 by POWER:26;
    hence thesis by ABSVALUE:def 1;
  end;
  suppose
A2: n is odd;

    (-1)|^n =(-1) to_power n by POWER:41
      .=-(1 to_power n) by A2,POWER:48
      .=-1 by POWER:26;
    then |.(-1)|^n.|=--1 by ABSVALUE:def 1;
    hence thesis;
  end;
end;
