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

theorem
  for Iac,Icb being non empty closed_interval Subset of REAL,
  Dac being Division of Iac, Dcb being Division of Icb st
  upper_bound(Iac) = lower_bound(Icb) &
  len Dcb = 1 &
  Dcb.1 = lower_bound(Icb) holds
  Dac (#) Dcb = Dac
  proof
    let Iac,Icb be non empty closed_interval Subset of REAL,
    Dac be Division of Iac, Dcb be Division of Icb;
    assume that
A1: upper_bound(Iac) = lower_bound(Icb) and
A2: len Dcb = 1 and
A3: Dcb.1 = lower_bound(Icb);
    len (Dcb/^1) = len Dcb - 1 by A2,RFINSEQ:def 1;
    then Dcb/^1 = {} by A2;
    then Dac ^ (Dcb/^1) = Dac by FINSEQ_1:34;
    hence thesis by A1,A3,Def4;
  end;
