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 being eventually-nondecreasing eventually-nonnegative
Real_Sequence , t being Real_Sequence st (for n holds (n mod 2 = 0 implies t.n
  = 1) & (n mod 2 = 1 implies t.n = n)) holds not t in Big_Theta(f)
proof
  let f be eventually-nondecreasing eventually-nonnegative Real_Sequence, t be
  Real_Sequence such that
A1: for n holds (n mod 2 = 0 implies t.n = 1) & (n mod 2 = 1 implies t.n = n);
A2: Big_Theta(f) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*f.n <= s.n & s.n <= c*f.n } by
ASYMPT_0:27;
  hereby
    consider N0 being Nat such that
A3: for n being Nat st n >= N0 holds f.n <= f.(n+1) by ASYMPT_0:def 6;
    assume t in Big_Theta(f);
    then consider s being Element of Funcs(NAT, REAL) such that
A4: s = t and
A5: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*f.n <= s.n & s
    .n <= c*f.n by A2;
    consider c,d,N such that
A6: c > 0 and
A7: d > 0 and
A8: for n st n >= N holds d*f.n <= s.n & s.n <= c*f.n by A5;
    set N1 = max([/c/d\] + 1, max(N, N0));
A9: N1 >= [/c/d\] + 1 by XXREAL_0:25;
A10: N1 is Integer by XXREAL_0:16;
A11: N1 >= max(N, N0) by XXREAL_0:25;
    max(N, N0) >= N0 by XXREAL_0:25;
    then
A12: N1 >= N0 by A11,XXREAL_0:2;
    max(N, N0) >= N by XXREAL_0:25;
    then
A13: N1 >= N by A11,XXREAL_0:2;
    reconsider N1 as Element of NAT by A11,A10,INT_1:3;
    thus contradiction
    proof
      per cases by NAT_D:12;
      suppose
A14:    N1 mod 2 = 1;
A15:    [/c/d\] >= c/d by INT_1:def 7;
        [/c/d\] + 1 > [/c/d\] + 0 by XREAL_1:8;
        then [/c/d\] + 1 > c/d by A15,XXREAL_0:2;
        then N1 > c/d by A9,XXREAL_0:2;
        then N1*c" > c"*(c/d) by A6,XREAL_1:68;
        then N1/c > (c"*c)*(1/d);
        then
A16:    N1/c > 1*(1/d) by A6,XCMPLX_0:def 7;
A17:    f.(N1+1) >= f.(N1) by A3,A12;
        s.N1 = N1 by A1,A4,A14;
        then N1 <= c*f.N1 by A8,A13;
        then N1/c <= f.N1 by A6,XREAL_1:79;
        then f.N1 > 1/d by A16,XXREAL_0:2;
        then f.(N1+1) > 1/d by A17,XXREAL_0:2;
        then
A18:    d*(1/d) < d*f.(N1+1) by A7,XREAL_1:68;
        N1+1 > N1+0 by XREAL_1:8;
        then
A19:    N1+1 > N by A13,XXREAL_0:2;
        (N1+1) mod 2 = (1+(1 mod 2)) mod 2 by A14,NAT_D:66
          .= (1+1) mod 2 by NAT_D:14
          .= 0 by NAT_D:25;
        then t.(N1+1) = 1 by A1;
        then d*f.(N1+1) <= 1 by A4,A8,A19;
        hence thesis by A7,A18,XCMPLX_1:106;
      end;
      suppose
A20:    N1 mod 2 = 0;
        then (N1+1) mod 2 = (0+(1 mod 2)) mod 2 by NAT_D:66
          .= (0+1) mod 2 by NAT_D:14
          .= 1 by NAT_D:14;
        then
A21:    s.(N1+1) = N1+1 by A1,A4;
A22:    [/c/d\] >= c/d by INT_1:def 7;
A23:    N1+1 > N1+0 by XREAL_1:8;
        then N1+1 > N0 by A12,XXREAL_0:2;
        then
A24:    f.((N1+1)+1) >= f.(N1+1) by A3;
        [/c/d\] + 1 > [/c/d\] + 0 by XREAL_1:8;
        then [/c/d\] + 1 > c/d by A22,XXREAL_0:2;
        then N1 > c/d by A9,XXREAL_0:2;
        then (N1+1) > c/d by A23,XXREAL_0:2;
        then (N1+1)*c" > c"*(c/d) by A6,XREAL_1:68;
        then (N1+1)/c > (c"*c)*(1/d);
        then
A25:    (N1+1)/c > 1*(1/d) by A6,XCMPLX_0:def 7;
        N1+1 > N by A13,A23,XXREAL_0:2;
        then N1+1 <= c*f.(N1+1) by A8,A21;
        then (N1+1)/c <= f.(N1+1) by A6,XREAL_1:79;
        then f.(N1+1) > 1/d by A25,XXREAL_0:2;
        then f.(N1+2) > 1/d by A24,XXREAL_0:2;
        then
A26:    d*(1/d) < d*f.(N1+2) by A7,XREAL_1:68;
        N1+2 > N1+0 by XREAL_1:8;
        then
A27:    N1+2 > N by A13,XXREAL_0:2;
        (N1+2) mod 2 = (0+(2 mod 2)) mod 2 by A20,NAT_D:66
          .= (0+0) mod 2 by NAT_D:25
          .= 0 by NAT_D:26;
        then t.(N1+2) = 1 by A1;
        then d*f.(N1+2) <= 1 by A4,A8,A27;
        hence thesis by A7,A26,XCMPLX_1:106;
      end;
    end;
  end;
end;
