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 :: Part 3
  ex s being eventually-positive Real_Sequence st s = seq_a^(2,1,1) &
  seq_a^(2,1,0) in Big_Theta(s)
proof
  reconsider g = seq_a^(2,1,1) as eventually-positive Real_Sequence;
  set f = seq_a^(2,1,0);
  take g;
  thus g = seq_a^(2,1,1);
A1: f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
A2: now
    let n;
    assume n >= 2;
A3: f.n = 2 to_power (1*n+0) by Def1;
A4: g.n = 2 to_power (1*n+1) by Def1;
    then (2 to_power -1)*g.n = 2 to_power ((-1)+(n+1)) by POWER:27
      .= f.n by A3;
    hence (2 to_power -1)*g.n <= f.n;
    n+0 <= n+1 by XREAL_1:7;
    hence f.n <= 1*g.n by A3,A4,PRE_FF:8;
  end;
A5: (2 to_power -1) > 0 by POWER:34;
  Big_Theta(g) = { 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*g.n <= s.n & s.n <= c*g.n } by
ASYMPT_0:27;
  hence thesis by A1,A5,A2;
end;
