theorem Th47:
  divset(D,1) = [.D.1,D.1.] implies D.1 = lower_bound I
  proof
    assume divset(D,1) = [.D.1,D.1.]; then
A1: lower_bound divset(D,1) = D.1 & upper_bound divset(D,1) = D.1
      by JORDAN5A:19;
A2: divset(D,1) = [.lower_bound I,D.1.] by COUSIN:50;
    rng D <> {};
    then 1 in dom D by FINSEQ_3:32;
    then D.1 in I by INTEGRA1:6;
    then D.1 in [. lower_bound I, upper_bound I.] by INTEGRA1:4;
    then lower_bound I <= D.1 <= upper_bound I by XXREAL_1:1;
    hence thesis by A1,A2,JORDAN5A:19;
  end;
