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 Th39:
  a>0 implies a #Z k > 0
proof
  assume
A1: a>0;
  per cases;
  suppose
    k>=0;
    then a #Z k = a |^ |.k.| by Def3;
    hence thesis by A1,Th6;
  end;
  suppose
A2: k<0;
A3: a |^ |.k.| > 0 by A1,Th6;
    a #Z k = (a |^ |.k.|)" by A2,Def3;
    hence thesis by A3;
  end;
end;
