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 Th74:
  a>0 implies a #R p = a #Q p
proof
  assume
A1: a>0;
  set s = seq_const p;
A2: lim s = s.0 by SEQ_4:26
    .= p by FUNCOP_1:7;
  reconsider s as Rational_Sequence;
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
      .= a #Q p by FUNCOP_1:7;
  end;
  a #Q p in REAL by XREAL_0:def 1;
  then
A4: a #Q s is constant by A3,VALUED_0:def 18;
  then lim (a #Q s) = (a #Q s).0 by SEQ_4:26
    .= a #Q p by A3;
  hence thesis by A1,A2,A4,Def6;
end;
