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

:: NOTE: The reverse implication is not true. Consider the case of
:: f = 1/n, c = 1. Then 1/n in Big_Oh(1+1/n), but 1+1/n is not in Big_Oh(1/n).
  for c being non negative Real, x,f being eventually-nonnegative
  Real_Sequence holds x in Big_Oh(f) implies x in Big_Oh(c+f)
proof
  let c be non negative Real, x,f be eventually-nonnegative Real_Sequence;
  assume x in Big_Oh(f);
  then consider t being Element of Funcs(NAT, REAL) such that
A1: x = t and
A2: ex c1,N st c1 > 0 & for n st n >= N holds t.n <= c1*f.n & t.n >= 0;
  consider c1,N such that
A3: c1 > 0 and
A4: for n st n >= N holds t.n <= c1*f.n & t.n >= 0 by A2;
  now
    let n;
    f.n+0 <= f.n+c by XREAL_1:7;
    then f.n <= (c+f).n by VALUED_1:2;
    then
A5: c1*f.n <= c1*(c+f).n by A3,XREAL_1:64;
    assume
A6: n >= N;
    then t.n <= c1*f.n by A4;
    hence t.n <= c1*(c+f).n & t.n >= 0 by A4,A6,A5,XXREAL_0:2;
  end;
  hence thesis by A1,A3;
end;
