reserve a,b,x,y for Real,
  i,k,n,m for Nat;
reserve p,w for Real;
reserve seq for Real_Sequence;

theorem Th3:
  seq is divergent_to+infty implies 2 to_power seq is divergent_to+infty
proof
  assume
A1: seq is divergent_to+infty;
  let r be Real;
  consider p being Nat such that
A2: p > r by SEQ_4:3;
  consider n such that
A3: for m st n <= m holds p < seq.m by A1;
  take n;
  let m;
  assume m >= n;
  then p < seq.m by A3;
  then
A4: 2 to_power p < 2 to_power (seq.m) by POWER:39;
  p < 2 to_power p by NEWTON:86;
  then p < 2 to_power (seq.m) by A4,XXREAL_0:2;
  then r < 2 to_power (seq.m) by A2,XXREAL_0:2;
  hence thesis by Def1;
end;
