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 A2: F is INT -valued by RELAT_1:def 19;
rng F c= COMPLEX by A1, MEMBERED:1;
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 = addint "**" F

hence addint "**" F = 0 by Def9, A2, BINOP_2:4
.= Sum F by Def9, A3, A4, BINOP_2:1 ;
:: thesis: verum

end;

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

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

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 Th59, A5, A6, A2; :: thesis: verum

end;

end;