let a1, a2, a3, a4, a5, a6, a7, b1, b2, b3, b4, b5, b6, b7 be Complex; :: thesis: <*a1,a2,a3,a4,a5,a6,a7*> + <*b1,b2,b3,b4,b5,b6,b7*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5),(a6 + b6),(a7 + b7)*>
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,a7*> = <*a1,a2,a3,a4,a5*> ^ <*a6,a7*> & <*b1,b2,b3,b4,b5,b6,b7*> = <*b1,b2,b3,b4,b5*> ^ <*b6,b7*> ) by AOFA_A00:def 5;
then <*a1,a2,a3,a4,a5,a6,a7*> + <*b1,b2,b3,b4,b5,b6,b7*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5)*> ^ <*(a6 + b6),(a7 + b7)*> by A1, FINSEQ940A;
hence <*a1,a2,a3,a4,a5,a6,a7*> + <*b1,b2,b3,b4,b5,b6,b7*> = <*(a1 + b1),(a2 + b2),(a3 + b3),(a4 + b4),(a5 + b5),(a6 + b6),(a7 + b7)*> by AOFA_A00:def 5; :: thesis: verum