
theorem Th15:
  for A being non empty Interval, b being ExtReal st
    ex a being ExtReal st a <= b & A = [.a,b.[ holds b = sup A
proof
  let A be non empty Interval, IT be ExtReal;
  given a being ExtReal such that
A1: a <= IT and
A2: A = [.a,IT.[;
  a <> IT by A2;
  then a < IT by A1,XXREAL_0:1;
  hence thesis by A2,XXREAL_2:31;
end;
