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

theorem
  for I1,I2,I being non empty closed_interval Subset of REAL st
    upper_bound I1 = lower_bound I2 & I = I1 \/ I2 holds
    lower_bound I = lower_bound I1 & upper_bound I = upper_bound I2
    proof
      let I1,I2,I being non empty closed_interval Subset of REAL;
      assume that
A1:   upper_bound I1 = lower_bound I2 and
A2:   I = I1 \/ I2;
A3:   I1 = [.lower_bound I1,upper_bound I1.] by INTEGRA1:4; then
A4:   lower_bound I1 <= upper_bound I1 by XXREAL_1:29;
A5:   I2 = [.lower_bound I2,upper_bound I2.] by INTEGRA1:4; then
A6:   lower_bound I2 <= upper_bound I2 by XXREAL_1:29;
A7:   I = [.lower_bound I1,upper_bound I2.] by A2,A3,A5,A1,A4,A6,XXREAL_1:165;
A8:   I = [.lower_bound I,upper_bound I.] by INTEGRA1:4;
      then lower_bound I <= upper_bound I by XXREAL_1:29;
      hence thesis by A7,A8,XXREAL_1:66;
    end;
