reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem
  for c being non negative Real, x,f being eventually-nonnegative
  Real_Sequence st ex e,N st e > 0 & for n st n >= N holds f.n >= e holds x in
  Big_Oh(c+f) implies x in Big_Oh(f)
proof
  let c be non negative Real, x,f be eventually-nonnegative Real_Sequence;
  given e,N0 such that
A1: e > 0 and
A2: for n st n >= N0 holds f.n >= e;
  assume x in Big_Oh(c+f);
  then consider t being Element of Funcs(NAT, REAL) such that
A3: x = t and
A4: ex d,N st d > 0 & for n st n >= N holds t.n <= d*(c+f).n & t.n >= 0;
  consider d,N1 such that
A5: d > 0 and
A6: for n st n >= N1 holds t.n <= d*(c+f).n & t.n >= 0 by A4;
  set b = max(2*d, (2*d*c)/e);
  2*d > 2*0 by A5,XREAL_1:68;
  then
A7: b > 0 by XXREAL_0:25;
  set N = max(N0, N1);
A8: N >= N1 by XXREAL_0:25;
A9: N >= N0 by XXREAL_0:25;
  now
    let n;
    assume
A10: n >= N;
    then
A11: n >= N1 by A8,XXREAL_0:2;
    then t.n <= d*(c+f).n by A6;
    then
A12: t.n <= d*(c+f.n) by VALUED_1:2;
A13: n >= N0 by A9,A10,XXREAL_0:2;
    thus t.n <= b*f.n
    proof
      per cases;
      suppose
        c >= f.n;
        then d*c >= d*f.n by A5,XREAL_1:64;
        then d*c + d*c >= d*c + d*f.n by XREAL_1:6;
        then t.n <= (2*(d*c))*1 by A12,XXREAL_0:2;
        then
A14:    t.n <= (2*(d*c))*((1/e)*e) by A1,XCMPLX_1:106;
        b*e >= ((2*d*c)/e)*e by A1,XREAL_1:64,XXREAL_0:25;
        then
A15:    t.n <= b*e by A14,XXREAL_0:2;
        b*f.n >= b*e by A2,A7,A13,XREAL_1:64;
        hence thesis by A15,XXREAL_0:2;
      end;
      suppose
        c < f.n;
        then d*c < d*f.n by A5,XREAL_1:68;
        then d*c + d*f.n < d*f.n + d*f.n by XREAL_1:6;
        then
A16:    t.n < 2*(d*f.n) by A12,XXREAL_0:2;
        f.n > 0 by A1,A2,A13;
        then b*f.n >= (2*d)*f.n by XREAL_1:64,XXREAL_0:25;
        hence thesis by A16,XXREAL_0:2;
      end;
    end;
    thus t.n >= 0 by A6,A11;
  end;
  hence thesis by A3,A7;
end;
