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 a,b being logbase Real, f,g being Real_Sequence st a > 1 & b > 1 &
  (for n st n > 0 holds f.n = log(a,n)) & (for n st n > 0 holds g.n = log(b,n))
  holds ex s,s1 being eventually-positive Real_Sequence st s = f & s1 = g &
  Big_Oh(s) = Big_Oh(s1)
proof
  let a,b be logbase Real, f,g be Real_Sequence such that
A1: a > 1 and
A2: b > 1 and
A3: for n st n > 0 holds f.n = log(a,n) and
A4: for n st n > 0 holds g.n = log(b,n);
  reconsider g as eventually-positive Real_Sequence by A2,A4,Lm2;
  reconsider f as eventually-positive Real_Sequence by A1,A3,Lm2;
  take f,g;
A5: a <> 1 by ASYMPT_0:def 1;
A6: b <> 1 by ASYMPT_0:def 1;
A7: b > 0 by ASYMPT_0:def 1;
A8: a > 0 by ASYMPT_0:def 1;
  now
    let x be object;
    hereby
      assume x in Big_Oh(f);
      then consider t being Element of Funcs(NAT, REAL) such that
A9:   x = t and
A10:  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
A11:  c > 0 and
A12:  for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A10;
A13:  now
        take N1 = N+1;
        let n;
        assume
A14:    n >= N1;
        then
A15:    f.n = log(a,n) by A3
          .= log(a,b)*log(b,n) by A8,A5,A7,A6,A14,POWER:56;
        N+1 > N+0 by XREAL_1:8;
        then
A16:    n > N by A14,XXREAL_0:2;
        then t.n <= c*f.n by A12;
        then t.n <= (c*log(a,b))*log(b,n) by A15;
        hence t.n <= (c*log(a,b))*g.n by A4,A14;
        thus t.n >= 0 by A12,A16;
      end;
      log(a,b) > log(a,1) by A1,A2,POWER:57;
      then log(a,b) > 0 by A8,A5,POWER:51;
      then c*log(a,b) > c*0 by A11,XREAL_1:68;
      hence x in Big_Oh(g) by A9,A13;
    end;
    assume x in Big_Oh(g);
    then consider t being Element of Funcs(NAT, REAL) such that
A17: x = t and
A18: 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
A19: c > 0 and
A20: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A18;
A21: now
      take N1 = N+1;
      let n;
      assume
A22:  n >= N1;
      then
A23:  g.n = log(b,n) by A4
        .= log(b,a)*log(a,n) by A8,A5,A7,A6,A22,POWER:56;
      N+1 > N+0 by XREAL_1:8;
      then
A24:  n > N by A22,XXREAL_0:2;
      then t.n <= c*g.n by A20;
      then t.n <= (c*log(b,a))*log(a,n) by A23;
      hence t.n <= (c*log(b,a))*f.n by A3,A22;
      thus t.n >= 0 by A20,A24;
    end;
    log(b,a) > log(b,1) by A1,A2,POWER:57;
    then log(b,a) > 0 by A7,A6,POWER:51;
    then c*log(b,a) > c*0 by A19,XREAL_1:68;
    hence x in Big_Oh(f) by A17,A21;
  end;
  hence thesis by TARSKI:2;
end;
