 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem
  for I being non empty closed_interval Subset of REAL, c being Real
  st c in I holds
  [.lower_bound I, c.] is non empty closed_interval Subset of REAL &
  [.c,upper_bound I.] is non empty closed_interval Subset of REAL &
  upper_bound [.lower_bound I,c.] = lower_bound [.c,upper_bound I.]
  proof
    let I be non empty closed_interval Subset of REAL,
    c be Real;
    assume
A1: c in I;
    consider a,b be Real such that
A2: a <= b and
A3: I = [.a,b.] by Th33;
A4: lower_bound(I) = a & upper_bound(I) = b by A2,A3,JORDAN5A:19;
A5: a <= c <= b by A1,A3,XXREAL_1:1;
    hence [.lower_bound(I),c.] is non empty closed_interval Subset of REAL &
    [.c,upper_bound(I).] is non empty closed_interval Subset of REAL
      by A4,XXREAL_1:30,MEASURE5:def 3;
    upper_bound [.lower_bound(I),c.] = c by A4,A5,JORDAN5A:19;
    hence upper_bound [.lower_bound(I),c.] = lower_bound [.c,upper_bound(I).]
      by A4,A5,JORDAN5A:19;
  end;
