let K be Field; :: thesis: for A1, A2, B1, B2 being Matrix of K holds <*A1,B1*> (+) <*A2,B2*> = <*(A1 + A2),(B1 + B2)*>
let A1, A2, B1, B2 be Matrix of K; :: thesis: <*A1,B1*> (+) <*A2,B2*> = <*(A1 + A2),(B1 + B2)*>
A1: len <*A2*> = 1 by FINSEQ_1:39;
len <*A1*> = 1 by FINSEQ_1:39;
hence <*A1,B1*> (+) <*A2,B2*> = (<*A1*> (+) <*A2*>) ^ (<*B1*> (+) <*B2*>) by A1, Th67
.= (<*A1*> (+) <*A2*>) ^ <*(B1 + B2)*> by Th69
.= <*(A1 + A2),(B1 + B2)*> by Th69 ;
:: thesis: verum