
theorem
  for a, b being Real st a <= b holds lower_bound [.a,b.] = a &
  upper_bound [.a,b.] = b
proof
  let a, b be Real;
  assume
A1: a <= b;
  set X = [.a,b.];
A2: a in { l where l is Real : a <= l & l <= b } by A1;
A3: for s be Real st 0 < s ex r being Real st r in X & r < a+ s
  proof
    let s be Real;
    assume
A4: 0 < s;
A5: a in X by A2,RCOMP_1:def 1;
    assume for r being Real st r in X holds r >= a+s;
    hence thesis by A4,A5,XREAL_1:29;
  end;
A6: b in { l1 where l1 is Real: a <= l1 & l1 <= b } by A1;
A7: for s be Real st 0 < s ex r being Real st r in X & b-s<r
  proof
    let s be Real;
    assume
A8: 0 < s;
A9: b in X by A6,RCOMP_1:def 1;
    assume for r being Real st r in X holds r <= b-s;
    hence thesis by A8,A9,XREAL_1:44;
  end;
A10: for r being Real st r in X holds a <= r
  proof
    let r be Real;
    assume r in X;
    then r in { l where l is Real: a <= l & l <= b } by RCOMP_1:def 1;
    then ex r1 being Real st r1 = r & a <= r1 & r1 <= b;
    hence thesis;
  end;
  b is UpperBound of X
  proof
    let r be ExtReal;
    assume r in X;
    then r in { l where l is Real: a <= l & l <= b } by RCOMP_1:def 1;
    then ex r1 being Real st r1 = r & a <= r1 & r1 <= b;
    hence thesis;
  end;
  then
A11: X is bounded_above;
 a is LowerBound of X
  proof
    let r be ExtReal;
    assume r in X;
    then r in { l where l is Real: a <= l & l <= b } by RCOMP_1:def 1;
    then ex r1 being Real st r1 = r & a <= r1 & r1 <= b;
    hence thesis;
  end;
  then
A12: X is bounded_below;
A13: for r being Real st r in X holds b >= r
  proof
    let r be Real;
    assume r in X;
    then r in { l where l is Real: a <= l & l <= b } by RCOMP_1:def 1;
    then ex r1 being Real st r1 = r & a <= r1 & r1 <= b;
    hence thesis;
  end;
  a in X by A2,RCOMP_1:def 1;
  hence thesis by A12,A11,A10,A3,A13,A7,SEQ_4:def 1,def 2;
end;
