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 Real_Sequence st
    (for n holds f.n = n mod 2) &
    (for n holds g.n = n+1 mod 2)
  ex s,s1 being eventually-nonnegative Real_Sequence
  st s = f & s1 = g & not s in Big_Oh(s1) & not s1 in Big_Oh(s)
proof
  let f,g be Real_Sequence such that
A1: for n holds f.n = n mod 2 and
A2: for n holds g.n = n+1 mod 2;
  g is eventually-nonnegative
  proof
    take 0;
    let n be Nat;
A3:  n in NAT by ORDINAL1:def 12;
    assume n >= 0;
A4: g.n = n+1 mod 2 by A2,A3;
    per cases by A4,NAT_D:12;
    suppose
      g.n = 0;
      hence thesis;
    end;
    suppose
      g.n = 1;
      hence thesis;
    end;
  end;
  then reconsider g as eventually-nonnegative Real_Sequence;
  f is eventually-nonnegative
  proof
    take 0;
    let n be Nat;
A5:  n in NAT by ORDINAL1:def 12;
    assume n >= 0;
A6: f.n = n mod 2 by A1,A5;
    per cases by A6,NAT_D:12;
    suppose
      f.n = 0;
      hence thesis;
    end;
    suppose
      f.n = 1;
      hence thesis;
    end;
  end;
  then reconsider f as eventually-nonnegative Real_Sequence;
A7: now
    assume g in Big_Oh(f);
    then consider t being Element of Funcs(NAT, REAL) such that
A8: t = g and
A9: 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
A10: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A9;
    ex n st n >= N & t.n > c*f.n
    proof
      per cases by NAT_D:12;
      suppose
A11:     N mod 2 = 0;
        then f.N = 0 by A1;
        then
A12:    c*f.N = 0;
        t.N = (N+1) mod 2 by A2,A8
          .= (0+(1 mod 2)) mod 2 by A11,NAT_D:66
          .= (0+1) mod 2 by NAT_D:14
          .= 1 by NAT_D:14;
        hence thesis by A12;
      end;
      suppose
A13:    N mod 2 = 1;
        f.(N+1) = (N+1) mod 2 by A1
          .= (1+(1 mod 2)) mod 2 by A13,NAT_D:66
          .= (1+1) mod 2 by NAT_D:14
          .= 0 by NAT_D:25;
        then
A14:    c*f.(N+1) = 0;
A15:    N+1 >= N by NAT_1:13;
        t.(N+1) = ((N+1)+1) mod 2 by A2,A8
          .= (N+(1+1)) mod 2
          .= (1+(2 mod 2)) mod 2 by A13,NAT_D:66
          .= (1+0) mod 2 by NAT_D:25
          .= 1 by NAT_D:14;
        hence thesis by A15,A14;
      end;
    end;
    hence contradiction by A10;
  end;
  take f,g;
  now
    assume f in Big_Oh(g);
    then consider t being Element of Funcs(NAT, REAL) such that
A16: t = f and
A17: 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
A18: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A17;
    ex n st n >= N & t.n > c*g.n
    proof
      per cases by NAT_D:12;
      suppose
A19:    N mod 2 = 0;
        g.(N+1) = ((N+1)+1) mod 2 by A2
          .= (N+(1+1)) mod 2
          .= (0+(2 mod 2)) mod 2 by A19,NAT_D:66
          .= (0+0) mod 2 by NAT_D:25
          .= 0 by NAT_D:26;
        then
A20:    c*g.(N+1) = 0;
A21:    N+1 >= N by NAT_1:13;
        t.(N+1) = (N+1) mod 2 by A1,A16
          .= (0+(1 mod 2)) mod 2 by A19,NAT_D:66
          .= (0+1) mod 2 by NAT_D:14
          .= 1 by NAT_D:14;
        hence thesis by A21,A20;
      end;
      suppose
A22:    N mod 2 = 1;
        g.N = (N+1) mod 2 by A2
          .= (1+(1 mod 2)) mod 2 by A22,NAT_D:66
          .= (1+1) mod 2 by NAT_D:14
          .= 0 by NAT_D:25;
        then
A23:    c*g.N = 0;
        t.N = 1 by A1,A16,A22;
        hence thesis by A23;
      end;
    end;
    hence contradiction by A18;
  end;
  hence thesis by A7;
end;
