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;
