let T be non empty reflexive transitive non void TA-structure ; :: thesis: for t being type of T
for v1, v2 being FinSequence of the adjectives of T
for i being natural number st i in dom v1 holds
(apply (v1 ^ v2),t) . i = (apply v1,t) . i
let t be type of T; :: thesis: for v1, v2 being FinSequence of the adjectives of T
for i being natural number st i in dom v1 holds
(apply (v1 ^ v2),t) . i = (apply v1,t) . i
let v1, v2 be FinSequence of the adjectives of T; :: thesis: for i being natural number st i in dom v1 holds
(apply (v1 ^ v2),t) . i = (apply v1,t) . i
set v = v1 ^ v2;
consider tt being FinSequence of the carrier of T, q being Element of T such that
A1: apply v1,t = tt ^ <*q*> by HILBERT2:4;
let i be natural number ; :: thesis: ( i in dom v1 implies (apply (v1 ^ v2),t) . i = (apply v1,t) . i )
A2: len (apply v1,t) = (len v1) + 1 by Def19;
assume A3: i in dom v1 ; :: thesis: (apply (v1 ^ v2),t) . i = (apply v1,t) . i
then A4: i >= 1 by FINSEQ_3:27;
len <*q*> = 1 by FINSEQ_1:56;
then (len v1) + 1 = (len tt) + 1 by A2, A1, FINSEQ_1:35;
then i <= len tt by A3, FINSEQ_3:27;
then A5: i in dom tt by A4, FINSEQ_3:27;
apply (v1 ^ v2),t = (apply v1,t) $^ (apply v2,(v1 ast t)) by Th35
.= tt ^ (apply v2,(v1 ast t)) by A1, REWRITE1:2 ;
then (apply (v1 ^ v2),t) . i = tt . i by A5, FINSEQ_1:def 7;
hence (apply (v1 ^ v2),t) . i = (apply v1,t) . i by A1, A5, FINSEQ_1:def 7; :: thesis: verum