
theorem LM000:
  for x be Nat ex m be Nat st x < 2 to_power m
  proof
    let x be Nat;
    per cases;
    suppose C1:x =0;
      take m=0;
      thus x < 2 to_power m by C1,POWER:24;
    end;
    suppose x <> 0; then
      0 < x; then
      P2: 2 to_power log (2,x) = x by POWER:def 3;
      X1: log (2,x) <= |. log (2,x) .| by COMPLEX1:76;
      0 < [/ |. log (2,x) .| \] + 1 by INT_1:32; then
      reconsider m = [/ |. log (2,x) .| \] + 1 as Nat
        by INT_1:3,ORDINAL1:def 12;
      P3:log (2,x) < m by X1,INT_1:32,XXREAL_0:2;
      take m;
      thus x < 2 to_power m by P2,P3,POWER:39;
    end;
  end;
