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

theorem Th2:
  for I be interval Subset of REAL st sup I in I holds sup I = upper_bound I
proof
    let I be interval Subset of REAL;
    assume sup I in I; then
    reconsider J=I as non empty right_end real-membered set by XXREAL_2:def 6;
    sup J = upper_bound J;
    hence sup I = upper_bound I;
end;
