reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th73:
  1 #R a = 1
proof
  consider s being Rational_Sequence such that
A1: s is convergent and
A2: a = lim s and
  for n holds s.n<=a by Th67;
  reconsider j = 1 as Element of REAL by NUMBERS:19;

A3: now
    let n be Nat;
    reconsider nn=n as Element of NAT by ORDINAL1:def 12;
    thus j #Q s .n = j #Q (s.nn) by Def5
      .= j by Th51;
  end;
  then j #Q s is constant by VALUED_0:def 18;
  then
A4: lim (1 #Q s) = 1 #Q s .0 by SEQ_4:26
    .= 1 by A3;
  1 #Q s is convergent by A1,Th69;
  hence thesis by A1,A2,A4,Def6;
end;
