let F be XFinSequence; :: thesis: ( F is rational-valued implies Sum F = addrat "**" F )
assume A1: F is rational-valued ; :: thesis: Sum F = addrat "**" F
then rng F c= RAT by VALUED_0:def 4;
then A2: F is RAT -valued by RELAT_1:def 19;
rng F c= COMPLEX by MEMBERED:1, A1;
then A3: F is COMPLEX -valued by RELAT_1:def 19;

per cases ( len F = 0 or len F >= 1 ) by NAT_1:14;

suppose A4: len F = 0 ; :: thesis: Sum F = addrat "**" F

hence addrat "**" F = 0 by BINOP_2:3, Def9, A2
.= Sum F by Def9, A3, A4, BINOP_2:1 ;
:: thesis: verum

end;

suppose A5: len F >= 1 ; :: thesis: Sum F = addrat "**" F

A6: RAT = RAT /\ COMPLEX by XBOOLE_1:28, MEMBERED:1;

now

let x, y be set ; :: thesis: ( x in RAT & y in RAT implies ( addrat . (x,y) = addcomplex . (x,y) & addrat . (x,y) in RAT ) )
assume ( x in RAT & y in RAT ) ; :: thesis: ( addrat . (x,y) = addcomplex . (x,y) & addrat . (x,y) in RAT )
then reconsider X = x, Y = y as Element of RAT ;
addrat . (x,y) = X + Y by BINOP_2:def 15;
hence ( addrat . (x,y) = addcomplex . (x,y) & addrat . (x,y) in RAT ) by BINOP_2:def 3; :: thesis: verum

end;

hence Sum F = addrat "**" F by Th60, A5, A6, A2; :: thesis: verum

end;

end;