let F be XFinSequence; :: thesis: ( F is integer-valued implies Sum F = addint "**" F )
assume A1: F is integer-valued ; :: thesis: Sum F = addint "**" F
then rng F c= INT by VALUED_0:def 5;
then A3: F is INT -valued by RELAT_1:def 19;
rng F c= COMPLEX by MEMBERED:1, A1;
then A4: F is COMPLEX -valued by RELAT_1:def 19;

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

suppose A5: len F = 0 ; :: thesis: Sum F = addint "**" F

hence addint "**" F = 0 by BINOP_2:4, Def9, A3
.= Sum F by Def9, A4, A5, BINOP_2:1 ;
:: thesis: verum

end;

suppose A6: len F >= 1 ; :: thesis: Sum F = addint "**" F

A7: INT = INT /\ COMPLEX by XBOOLE_1:28, MEMBERED:1;

now

let x, y be set ; :: thesis: ( x in INT & y in INT implies ( addint . (x,y) = addcomplex . (x,y) & addint . (x,y) in INT ) )
assume ( x in INT & y in INT ) ; :: thesis: ( addint . (x,y) = addcomplex . (x,y) & addint . (x,y) in INT )
then reconsider X = x, Y = y as Element of INT ;
addint . (x,y) = X + Y by BINOP_2:def 20;
hence ( addint . (x,y) = addcomplex . (x,y) & addint . (x,y) in INT ) by BINOP_2:def 3; :: thesis: verum

end;

hence Sum F = addint "**" F by Th60, A6, A7, A3; :: thesis: verum

end;

end;