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 Th71:
  a > 0 implies a #R 0 = 1
proof
  set s = seq_const 0;
  reconsider s as Rational_Sequence;
  assume
A1: a > 0;
  s.0 = 0;
  then
A2: lim s = 0 by SEQ_4:26;
   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 a #Q s .n = a #Q (s.nn) by Def5
      .= j by Th47;
  end;
  then
A4: a #Q s is constant by VALUED_0:def 18;
  a #Q s .0 = 1 by A3;
  then lim (a #Q s) = 1 by A4,SEQ_4:26;
  hence thesis by A1,A2,A4,Def6;
end;
