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 f,g being eventually-nonnegative Real_Sequence holds
    Big_Oh(f) = Big_Oh(g) iff f in Big_Theta(g)
proof
  let f,g be eventually-nonnegative Real_Sequence;
  hereby
    assume
A1: Big_Oh(f) = Big_Oh(g);
    then g in Big_Oh(f) by ASYMPT_0:10;
    then
A2: f in Big_Omega(g) by ASYMPT_0:19;
    f in Big_Oh(g) by A1,ASYMPT_0:10;
    hence f in Big_Theta(g) by A2,XBOOLE_0:def 4;
  end;
  assume
A3: f in Big_Theta(g);
  now
    let x be object;
    hereby
      assume x in Big_Oh(f);
      then consider t being Element of Funcs(NAT, REAL) such that
A4:   x = t and
A5:   ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
      consider c,N such that
      c > 0 and
A6:   for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A5;
      now
         reconsider N as Nat;
        take N;
        let n be Nat;
A7:      n in NAT by ORDINAL1:def 12;
        assume n >= N;
        hence t.n >= 0 by A6,A7;
      end;
      then
A8:   t is eventually-nonnegative;
A9:   f in Big_Oh(g) by A3,XBOOLE_0:def 4;
      t in Big_Oh(f) by A5;
      hence x in Big_Oh(g) by A4,A8,A9,ASYMPT_0:12;
    end;
    assume x in Big_Oh(g);
    then consider t being Element of Funcs(NAT, REAL) such that
A10: x = t and
A11: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n >= 0;
    consider c,N such that
    c > 0 and
A12: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A11;
    now
       reconsider N as Nat;
      take N;
      let n be Nat;
A13:      n in NAT by ORDINAL1:def 12;
      assume n >= N;
      hence t.n >= 0 by A12,A13;
    end;
    then
A14: t is eventually-nonnegative;
    f in Big_Omega(g) by A3,XBOOLE_0:def 4;
    then
A15: g in Big_Oh(f) by ASYMPT_0:19;
    t in Big_Oh(g) by A11;
    hence x in Big_Oh(f) by A10,A14,A15,ASYMPT_0:12;
  end;
  hence thesis by TARSKI:2;
end;
