let a be real number ; :: thesis: ( a > 0 implies a #R 0 = 1 )
assume A1: a > 0 ; :: thesis: a #R 0 = 1
reconsider s = NAT --> 0 as Real_Sequence by FUNCOP_1:57;
for n being Element of NAT holds s . n is Rational ;
then reconsider s = s as Rational_Sequence by Def6;
s . 0 = 0 by FUNCOP_1:13;
then A3: lim s = 0 by SEQ_4:41;

A4: now

let n be Nat; :: thesis: (a #Q s) . n = 1
reconsider nn = n as Element of NAT by ORDINAL1:def 13;
thus (a #Q s) . n = a #Q (s . nn) by Def7
.= 1 by A1, Th58, FUNCOP_1:13 ; :: thesis: verum

end;

then A5: a #Q s is constant by VALUED_0:def 18;
then A6: a #Q s is convergent ;
(a #Q s) . 0 = 1 by A4;
then lim (a #Q s) = 1 by A5, SEQ_4:41;
hence a #R 0 = 1 by A1, A3, A6, Def8; :: thesis: verum