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

theorem Th12:
  for a being R_eal st for r being Real holds r < a
      holds a = +infty
proof
  let a being R_eal;
  assume
A1: for r being Real holds r < a;
  assume not a = +infty;
then  a < +infty by XXREAL_0:4;
  then consider b being R_eal such that
A2: a < b and b < +infty and
A3: b in REAL by MEASURE5:2;
  reconsider b as Real by A3;
 a <= b by A2;
  hence contradiction by A1;
end;
