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 Th11:
  for f,g being eventually-nonnegative Real_Sequence st f in
  Big_Oh(g) holds Big_Oh(f) c= Big_Oh(g)
proof
  let f,g be eventually-nonnegative Real_Sequence;
  assume f in Big_Oh(g);
  then consider t being Element of Funcs(NAT, REAL) such that
A1: t = f and
A2: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n>=0;
  consider ct being Real, Nt being Element of NAT such that
  ct > 0 and
A3: for n st n >= Nt holds t.n <= ct*g.n & t.n >= 0 by A2;
  consider Ng being Nat such that
A4: for n being Nat st n >= Ng holds g.n >= 0 by Def2;
  let x be object;
  assume x in Big_Oh(f);
  then consider s being Element of Funcs(NAT, REAL) such that
A5: s = x and
A6: ex c,N st c > 0 & for n st n >= N holds s.n <= c*f.n & s.n >= 0;
  consider cs being Real, Ns being Element of NAT such that
A7: cs > 0 and
A8: for n st n >= Ns holds s.n <= cs*f.n & s.n >= 0 by A6;
  now
    take c = max(cs*ct,max(cs,ct));
    c >= max(cs,ct) by XXREAL_0:25;
    hence c > 0 by A7,XXREAL_0:25;
     reconsider N = max(max(Ns, Nt), Ng) as Element of NAT
          by ORDINAL1:def 12;
    take N;
    let n;
    assume
A9: n >= N;
A10: N >= max(Ns,Nt) by XXREAL_0:25;
    max(Ns,Nt) >= Nt by XXREAL_0:25;
    then N >= Nt by A10,XXREAL_0:2;
    then n >= Nt by A9,XXREAL_0:2;
    then t.n <= ct*g.n by A3;
    then
A11: cs*t.n <= cs*(ct*g.n) by A7,XREAL_1:64;
    N >= Ng by XXREAL_0:25;
    then n >= Ng by A9,XXREAL_0:2;
    then g.n >= 0 by A4;
    then cs*ct*g.n <= c*g.n by XREAL_1:64,XXREAL_0:25;
    then
A12: cs*t.n <= c*g.n by A11,XXREAL_0:2;
    max(Ns,Nt) >= Ns by XXREAL_0:25;
    then N >= Ns by A10,XXREAL_0:2;
    then
A13: n >= Ns by A9,XXREAL_0:2;
    then s.n <= cs*f.n by A8;
    hence s.n <= c*g.n by A1,A12,XXREAL_0:2;
    thus s.n >= 0 by A8,A13;
  end;
  hence thesis by A5;
end;
