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 Th40:
  (a*b) #Z k = a #Z k * b #Z k
proof
  per cases;
  suppose
A1: k>=0;
    hence (a*b) #Z k = (a*b) |^ |.k.| by Def3
      .= a |^ |.k.| * b |^ |.k.| by NEWTON:7
      .= a #Z k * b |^ |.k.| by A1,Def3
      .= a #Z k * b #Z k by A1,Def3;
  end;
  suppose
A2: k<0;
    hence (a*b) #Z k = ((a*b) |^ |.k.|)" by Def3
      .= (a |^ |.k.| * b |^ |.k.|)" by NEWTON:7
      .= (a |^ |.k.|)" * (b |^ |.k.|)" by XCMPLX_1:204
      .= a #Z k * (b |^ |.k.|)" by A2,Def3
      .= a #Z k * b #Z k by A2,Def3;
  end;
end;
