let p, q be Element of REAL 3; :: thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of ,
for t1, t2 being Real st p = (VFunc f1,f2,f3) . t1 & q = (VFunc g1,g2,g3) . t2 holds
|(p,q)| = (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (g3 . t2))
let f1, f2, f3, g1, g2, g3 be PartFunc of ,; :: thesis: for t1, t2 being Real st p = (VFunc f1,f2,f3) . t1 & q = (VFunc g1,g2,g3) . t2 holds
|(p,q)| = (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (g3 . t2))
let t1, t2 be Real; :: thesis: ( p = (VFunc f1,f2,f3) . t1 & q = (VFunc g1,g2,g3) . t2 implies |(p,q)| = (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (g3 . t2)) )
assume A1: ( p = (VFunc f1,f2,f3) . t1 & q = (VFunc g1,g2,g3) . t2 ) ; :: thesis: |(p,q)| = (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (g3 . t2))
|(p,q)| = (((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3)) by Lm8
.= (((f1 . t1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3)) by A1, Th34
.= (((f1 . t1) * (q . 1)) + ((f2 . t1) * (q . 2))) + ((p . 3) * (q . 3)) by A1, Th34
.= (((f1 . t1) * (q . 1)) + ((f2 . t1) * (q . 2))) + ((f3 . t1) * (q . 3)) by A1, Th34
.= (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (q . 2))) + ((f3 . t1) * (q . 3)) by A1, Th34
.= (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (q . 3)) by A1, Th34
.= (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (g3 . t2)) by A1, Th34 ;
hence |(p,q)| = (((f1 . t1) * (g1 . t2)) + ((f2 . t1) * (g2 . t2))) + ((f3 . t1) * (g3 . t2)) ; :: thesis: verum