let cF be complex-valued XFinSequence; :: thesis: ( cF is INT -valued implies Product cF = multint "**" cF )
assume A1: cF is INT -valued ; :: thesis: Product cF = multint "**" cF

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

suppose A2: len cF = 0 ; :: thesis: Product cF = multint "**" cF

hence multint "**" cF = the_unity_wrt multcomplex by BINOP_2:6, BINOP_2:9, AFINSQ_2:def 8, A1
.= Product cF by AFINSQ_2:def 8, A2 ;
:: thesis: verum

end;

suppose A3: len cF >= 1 ; :: thesis: Product cF = multint "**" cF

A4: INT = INT /\ COMPLEX by MEMBERED:1, XBOOLE_1:28;

now :: thesis: for x, y being object st x in INT & y in INT holds
( multint . (x,y) = multcomplex . (x,y) & multint . (x,y) in INT )

let x, y be object ; :: thesis: ( x in INT & y in INT implies ( multint . (x,y) = multcomplex . (x,y) & multint . (x,y) in INT ) )
assume ( x in INT & y in INT ) ; :: thesis: ( multint . (x,y) = multcomplex . (x,y) & multint . (x,y) in INT )
then reconsider X = x, Y = y as Element of INT ;
multint . (x,y) = X * Y by BINOP_2:def 22;
hence ( multint . (x,y) = multcomplex . (x,y) & multint . (x,y) in INT ) by BINOP_2:def 5, INT_1:def 2; :: thesis: verum

end;

hence Product cF = multint "**" cF by AFINSQ_2:47, A3, A4, A1; :: thesis: verum

end;

end;