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

assume A1: cF is INT -valued ; :: thesis: Product cF = multint "**" cF

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

end;

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;.= Product cF by AFINSQ_2:def 8, A2 ;

:: thesis: verum

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

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

end;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 )

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