reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th41:
  a #Z (-k) = 1/a #Z k
proof
  per cases;
  suppose
A1: k>0;
    then -k<-0;
    hence a #Z (-k) = (a |^ |.-k.|)" by Def3
      .= (a |^ |.k.|)" by COMPLEX1:52
      .= 1 / a |^ |.k.|
      .= 1 / a #Z k by A1,Def3;
  end;
  suppose
A2: k=0;
    hence a #Z (-k) = 1/1 by Th34
      .= 1/a GeoSeq.0 by Th3
      .= 1/a |^ 0 by Def1
      .= 1/a #Z k by A2,Th36;
  end;
  suppose
A3: k<0;
    then a #Z k = (a |^ |.k.|)" by Def3
      .= 1 / a |^ |.k.|
      .= 1 / a |^ |.-k.| by COMPLEX1:52
      .= 1 / a #Z (-k) by A3,Def3;
    hence thesis;
  end;
end;
