
theorem Th22:
  for a,b being Real st 0 <= a & 1 < b - a
   ex n being Nat st a < n & n < b
proof
  let a,b be Real;
  assume that
A1: 0 <= a and
A2: 1 < b - a;
  a < [\a/]+1 by INT_1:29;
  then reconsider n = [\a/]+1 as Element of NAT by A1,INT_1:3;
  take n;
  thus a < n by INT_1:29;
  [\a/] <= a by INT_1:def 6;
  then
A3: [\a/]+1 <= a+1 by XREAL_1:6;
  1+a < b by A2,XREAL_1:20;
  hence thesis by A3,XXREAL_0:2;
end;
