reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th26:
  for a, b being Real st a < b ex p being irrational Real st a < p & p < b
proof
  set x = frac number_e;
  let a, b be Real;
A1: x < 1 by INT_1:43;
  assume a < b;
  then consider p1, p2 being Rational such that
A2: a < p1 and
A3: p1 < p2 and
A4: p2 < b by Th25;
  set y = (1 - x) * p1 + x * p2;
A5: 0 < x by INT_1:43;
  then y < p2 by A3,A1,XREAL_1:178;
  then
A6: y < b by A4,XXREAL_0:2;
  y > p1 by A3,A5,A1,XREAL_1:177;
  then
A7: y > a by A2,XXREAL_0:2;
A8: y = p1 + x * (p2 - p1);
  p2 - p1 <> 0 by A3;
  hence thesis by A8,A6,A7;
end;
