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
  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,Th5;
  end;
  suppose
    k<0;
    then a #Z k = (a |^ |.k.|)" by Def3;
    hence thesis by A1,Th5,XCMPLX_1:202;
  end;
end;
