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

theorem
  for s1,s2 being non empty increasing FinSequence of REAL,
      r being Real st s1.len s1 < r < s2.1
  holds s1 ^ <*r*> ^ s2 is non empty increasing FinSequence of REAL
  proof
    let s1,s2 be non empty increasing FinSequence of REAL,
        r be Real;
    assume
A1: s1.len s1 < r < s2.1;
    then reconsider s = s1 ^ <*r*> as
      non empty increasing FinSequence of REAL by Th38;
    s.len s = s.(len s1 + 1) by FINSEQ_2:16
           .= r by FINSEQ_1:42;
    hence thesis by A1,Th1;
  end;
