let a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6 be Complex; :: thesis: <*a1,a2,a3,a4,a5,a6*> + <*b1,b2,b3,b4,b5,b6*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5),(a6 + b6)*>
A1: <*a1,a2,a3,a4,a5*> + <*b1,b2,b3,b4,b5*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5)*> by A5B5;
( <*a1,a2,a3,a4,a5,a6*> = <*a1,a2,a3,a4,a5*> ^ <*a6*> & <*b1,b2,b3,b4,b5,b6*> = <*b1,b2,b3,b4,b5*> ^ <*b6*> ) by AOFA_A00:def 4;
then <*a1,a2,a3,a4,a5,a6*> + <*b1,b2,b3,b4,b5,b6*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5)*> ^ <*(a6 + b6)*> by A1, FINSEQ_9:40
.= <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5),(a6 + b6)*> by AOFA_A00:def 4 ;
hence <*a1,a2,a3,a4,a5,a6*> + <*b1,b2,b3,b4,b5,b6*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5),(a6 + b6)*> ; :: thesis: verum