let a be real number ; :: thesis: for p being Rational st a > 0 holds
a #R p = a #Q p
let p be Rational; :: thesis: ( a > 0 implies a #R p = a #Q p )
assume A1: a > 0 ; :: thesis: a #R p = a #Q p
p in REAL by XREAL_0:def 1;
then reconsider s = NAT --> p as Real_Sequence by FUNCOP_1:45;
A2: lim s = s . 0 by SEQ_4:26
.= p by FUNCOP_1:7 ;
reconsider s = s as Rational_Sequence ;

A3: now

let n be Nat; :: thesis: (a #Q s) . n = a #Q p
reconsider nn = n as Element of NAT by ORDINAL1:def 12;
thus (a #Q s) . n = a #Q (s . nn) by Def7
.= a #Q p by FUNCOP_1:7 ; :: thesis: verum

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 a #R p = a #Q p by A1, A2, A4, Def8; :: thesis: verum