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

theorem Th11:
  for r,s being Real st for n being Element of NAT
  holds r-1/(n+1) <= s holds r <= s
proof
  let r,s be Real;
  assume
A1: for n being Element of NAT holds r-1/(n+1) <= s;
  assume r > s;
  then consider n being Element of NAT such that
A2: 1/(n+1) < r - s by Th10;
 s + 1/(n+1) < r by A2,XREAL_1:20;
then  s < r - 1/(n+1) by XREAL_1:20;
  hence contradiction by A1;
end;
