let F be XFinSequence; :: thesis: ( F is INT -valued implies Sum F = addint "**" F )
assume A1: F is INT -valued ; :: thesis: Sum F = addint "**" F
rng F c= COMPLEX by A1, MEMBERED:1;
then A2: F is COMPLEX -valued by RELAT_1:def 19;
per cases ( len F = 0 or len F >= 1 ) by NAT_1:14;
suppose A3: len F = 0 ; :: thesis: Sum F = addint "**" F
hence addint "**" F = 0 by Def8, A1, BINOP_2:4
.= Sum F by Def8, A2, A3, BINOP_2:1 ;
:: thesis: verum
end;
suppose A4: len F >= 1 ; :: thesis: Sum F = addint "**" F
A5: INT = INT /\ COMPLEX by MEMBERED:1, XBOOLE_1:28;
now :: thesis: for x, y being object st x in INT & y in INT holds
( addint . (x,y) = addcomplex . (x,y) & addint . (x,y) in INT )
let x, y be object ; :: 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, INT_1:def 2; :: thesis: verum
end;
hence Sum F = addint "**" F by Th46, A4, A5, A1; :: thesis: verum
end;
end;