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 Th4:
  for f,g being eventually-nonnegative Real_Sequence holds Big_Oh(f
  ) c= Big_Oh(g) & not Big_Oh(f) = Big_Oh(g) iff f in Big_Oh(g) & not f in
  Big_Omega(g)
proof
  let f,g be eventually-nonnegative Real_Sequence;
  hereby
    assume that
A1: Big_Oh(f) c= Big_Oh(g) and
A2: not Big_Oh(f) = Big_Oh(g);
A3: f in Big_Oh(f) by ASYMPT_0:10;
    now
      assume f in Big_Omega(g);
      then g in Big_Oh(f) by ASYMPT_0:19;
      hence contradiction by A1,A2,A3,Lm5;
    end;
    hence f in Big_Oh(g) & not f in Big_Omega(g) by A1,A3;
  end;
  assume that
A4: f in Big_Oh(g) and
A5: not f in Big_Omega(g);
  now
    let x be object;
    assume x in Big_Oh(f);
    then consider t being Element of Funcs(NAT, REAL) such that
A6: x = t and
A7: 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
A8: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A7;
    now
       reconsider N as Nat;
      take N;
      let n be Nat;
A9:  n in NAT by ORDINAL1:def 12;
      assume n >= N;
      hence t.n >= 0 by A8,A9;
    end;
    then
A10: t is eventually-nonnegative;
    t in Big_Oh(f) by A7;
    hence x in Big_Oh(g) by A4,A6,A10,ASYMPT_0:12;
  end;
  hence Big_Oh(f) c= Big_Oh(g) by TARSKI:def 3;
  assume Big_Oh(f) = Big_Oh(g);
  then g in Big_Oh(f) by Lm5;
  hence contradiction by A5,ASYMPT_0:19;
end;
