reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for f be real-valued FinSequence,
      g be nonnegative-yielding real-valued FinSequence st
  (for x be Nat holds f.x >= g.x) holds f is nonnegative-yielding
proof
  let f be real-valued FinSequence,
  g be nonnegative-yielding real-valued FinSequence
    such that
  A1: for x be Nat holds f.x >= g.x;
  for r be Real st r in rng f holds r >= 0
  proof
    let r be Real such that
    B0: r in rng f;
    consider k be object such that
    B1: k in dom f & r = f.k by B0,FUNCT_1:def 3;
    reconsider k as Nat by B1;
    g.k >= 0;
    hence thesis by B1,A1;
  end;
  hence thesis by PARTFUN3:def 4;
end;
