reserve k for Element of NAT;
reserve r,r1 for Real;
reserve i for Integer;
reserve q for Rational;

theorem Th10:
  for r,s being Real st r < s holds
  ex n being Element of NAT st 1/(n+1) < s-r
proof
  let r,s be Real;
  assume r < s;
then  s-r > 0 by XREAL_1:50;
  then consider t being Real such that
A1: 0 < t and
A2: t < s-r by XREAL_1:5;
  reconsider t as Real;
A3: 1/t > 0 by A1,XREAL_1:139;
A4: [/ 1/t \] + 1 > 1/t by INT_1:32;
  set n = [/ 1/t \];
  reconsider n as Element of NAT by A3,A4,INT_1:3,7;
 (n+1)*t >= 1 by A1,A4,XREAL_1:81;
then  1/(n+1) <= t by XREAL_1:79;
then  1/(n+1) < s-r by A2,XXREAL_0:2;
  hence thesis;
end;
