 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem Th1:
  for I be Interval st inf I in I holds inf I = lower_bound I
proof
    let I be Interval;
    assume inf I in I; then
    reconsider J=I as non empty left_end real-membered set by XXREAL_2:def 5;
    inf J = lower_bound J;
    hence inf I = lower_bound I;
end;
