reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

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;
