
theorem
  for A being non empty Subset of ExtREAL holds for a,b being R_eal st A
  c= [.a,b.] holds a <= inf A & sup A <= b
proof
  let A be non empty Subset of ExtREAL;
  let a,b be R_eal;
  assume
A1: A c= [.a,b.];
  then reconsider B = [.a,b.] as non empty Subset of ExtREAL by MEMBERED:2;
  for x being ExtReal st x in B holds x <= b by XXREAL_1:1;
  then b is UpperBound of B by XXREAL_2:def 1;
  then
A2: b is UpperBound of A by A1,XXREAL_2:6;
  for x being ExtReal holds (x in B implies a <= x) by XXREAL_1:1;
  then a is LowerBound of B by XXREAL_2:def 2;
  then a is LowerBound of A by A1,XXREAL_2:5;
  hence thesis by A2,XXREAL_2:def 3,def 4;
end;
