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 positive Real st a < b
   holds not seq_a^(b,1,0) in Big_Oh(seq_a^(a,1,0))
proof
  let a,b be positive Real such that
A1: a < b;
  set g = seq_a^(a,1,0);
  set f = seq_a^(b,1,0);
  hereby
    set d = (log(2,b) - log(2,a));
    assume f in Big_Oh(g);
    then consider s being Element of Funcs(NAT, REAL) such that
A2: s = f and
A3: ex c,N st c > 0 & for n st n >= N holds s.n <= c*g.n & s.n >= 0;
    consider c,N such that
A4: c > 0 and
A5: for n st n >= N holds s.n <= c*g.n & s.n >= 0 by A3;
    set N0 = [/log(2,c) / d\];
    set N1 = max( N, N0 );
A6: N1 >= N by XXREAL_0:25;
A7: N1 = N or N1 = N0 by XXREAL_0:16;
A8: N1 >= N0 by XXREAL_0:25;
    reconsider N1 as Element of NAT by A6,A7,INT_1:3;
    set n = N1 + 1;
    set e = (2 to_power (n*log(2,a)));
A9: e > 0 by POWER:34;
A10: N0 >= log(2,c) / d by INT_1:def 7;
    log(2,b) > log(2,a) + 0 by A1,POWER:57;
    then
A11: d > 0 by XREAL_1:20;
A12: N1 + 1 > N1 + 0 by XREAL_1:8;
    then n > N0 by A8,XXREAL_0:2;
    then n > log(2,c) / d by A10,XXREAL_0:2;
    then n*d > (log(2,c) / d)*d by A11,XREAL_1:68;
    then n*d > log(2,c) by A11,XCMPLX_1:87;
    then 2 to_power (n*d) > 2 to_power log(2,c) by POWER:39;
    then 2 to_power (n*log(2,b) - n*log(2,a)) > c by A4,POWER:def 3;
    then (2 to_power (n*log(2,b))) / e > c by POWER:29;
    then ((2 to_power (n*log(2,b)))/e)*e > c*e by A9,XREAL_1:68;
    then 2 to_power (n*log(2,b)) > c*e by A9,XCMPLX_1:87;
    then b to_power n > c*(2 to_power (n*log(2,a))) by Lm3;
    then
A13: b to_power n > c*(a to_power n) by Lm3;
    n > N by A6,A12,XXREAL_0:2;
    then f.n <= c*g.n by A2,A5;
    then b to_power (1*n + 0) <= c*g.n by Def1;
    hence contradiction by A13,Def1;
  end;
end;
