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 Th42:
  (1/a) #Z k = 1/a #Z k
proof
  per cases;
  suppose
A1: k>=0;
    hence (1/a) #Z k = (1/a) |^ |.k.| by Def3
      .= 1 / a |^ |.k.| by Th7
      .= 1 / a #Z k by A1,Def3;
  end;
  suppose
A2: k<0;
    hence (1/a) #Z k = ((1/a) |^ |.k.|)" by Def3
      .= (1 / a |^ |.k.|)" by Th7
      .= (a |^ |.k.|)""
      .= (a #Z k)" by A2,Def3
      .= 1/a #Z k;
  end;
end;
