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

theorem
  for I1, I2 being non empty closed_interval Subset of REAL
  st upper_bound I1 = lower_bound I2 holds
  ex a,b,c being Real st a <= c <= b & I1 = [.a,c.] & I2 = [.c,b.]
  proof
    let I1, I2 be non empty closed_interval Subset of REAL;
    assume
A1: upper_bound I1 = lower_bound I2;
    consider a1,b1 be Real such that
A2: a1 <= b1 and
A3: I1 = [.a1,b1.] by Th33;
    consider a2,b2 be Real such that
A4: a2 <= b2 and
A5: I2 = [.a2,b2.] by Th33;
A6: upper_bound I1 = b1 by A2,A3,JORDAN5A:19;
    lower_bound I2 = a2 by A4,A5,JORDAN5A:19;
    hence thesis by A1,A2,A4,A3,A5,A6;
  end;
