theorem Th17:
  seq is nonnegative implies seq ^\k is nonnegative
proof
  assume
A1: seq is nonnegative;
  for n holds (seq ^\k).n >= 0
  proof
    let n;
    (seq ^\k).n = seq.(n+k) by NAT_1:def 3;
    hence thesis by A1;
  end;
  hence thesis;
end;
