
theorem Th6:
  for m,n being Element of NAT st m>0 ex i being Element of NAT st
  (for k2 being Element of NAT st k2<i holds m*(2|^k2)<=n) & m*(2|^i)>n
proof
  let m,n be Element of NAT;
  defpred P[Nat] means m*(2|^$1)> n;
  n+1-1=n;
  then
A1: n+1-'1=n by XREAL_0:def 2;
  assume
A2: m>0;
  then m>=0+1 by NAT_1:13;
  then
A3: m*n>=1*n by XREAL_1:64;
  2|^(n+1-'1)>n+1-'1 by NEWTON:86;
  then m*(2|^(n+1-'1))>m*(n+1-'1) by A2,XREAL_1:68;
  then m*(2|^(n+1-'1))> n by A1,A3,XXREAL_0:2;
  then
A4: ex k being Nat st P[k];
  ex k being Nat st P[k] & for j being Nat st P[j] holds k<=j from NAT_1:
  sch 5(A4);
  then consider k being Nat such that
A5: P[k] and
A6: for j being Nat st P[j] holds k<=j;
  k in NAT & for k2 being Element of NAT st k2<k holds m*(2|^(k2))<=n by A6,
ORDINAL1:def 12;
  hence thesis by A5;
end;
