theorem Th45:
  for X being non empty real-membered set, r for t st for s st s
  in X holds s <= t holds upper_bound X <= t
proof
  let X be non empty real-membered set, r;
  set r = upper_bound X;
  let t;
  assume
A1: for s st s in X holds s <= t;
  set s = r-t;
  assume r > t; then
A2: s > 0 by XREAL_1:50;
  X is bounded_above proof take t;
    let s be ExtReal;
    thus thesis by A1;
   end;
  then ex t9 be Real st t9 in X & r-s < t9 by A2,Def1;
  hence contradiction by A1;
end;
