reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem
  for x being set, f being eventually-nonnegative Real_Sequence st x in
  Big_Omega(f) holds x is eventually-nonnegative Real_Sequence
proof
  let t be set, f be eventually-nonnegative Real_Sequence;
  assume t in Big_Omega(f);
  then consider s being Element of Funcs(NAT, REAL) such that
A1: s = t and
A2: ex d,N st d > 0 & for n st n >= N holds s.n >= d*f.n & s.n >= 0;
  reconsider t9 = t as Real_Sequence by A1;
  consider d,N such that
  d > 0 and
A3: for n st n >= N holds s.n >= d*f.n & s.n >= 0 by A2;
  now
     reconsider N as Nat;
    take N;
    let n be Nat;
A4:  n in NAT by ORDINAL1:def 12;
    assume n >= N;
    hence t9.n >= 0 by A1,A3,A4;
  end;
  hence thesis by Def2;
end;
