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;
A1: ( len (apply v1,t) = (len v1) + 1 & len (apply v2,(v1 ast t)) = (len v2) + 1 ) by Def19;
consider tt being FinSequence of the carrier of T, q being Element of T such that
A2: apply v1,t = tt ^ <*q*> by HILBERT2:4;
A3: apply (v1 ^ v2),t = (apply v1,t) $^ (apply v2,(v1 ast t)) by Th35
.= tt ^ (apply v2,(v1 ast t)) by A2, REWRITE1:2 ;
len <*q*> = 1 by FINSEQ_1:56;
then A4: (len v1) + 1 = (len tt) + 1 by A1, A2, FINSEQ_1:35;
let i be natural number ; :: thesis: ( i in dom v1 implies (apply (v1 ^ v2),t) . i = (apply v1,t) . i )
assume i in dom v1 ; :: thesis: (apply (v1 ^ v2),t) . i = (apply v1,t) . i
then ( i >= 1 & i <= len tt & i is Element of NAT ) by A4, FINSEQ_3:27;
then i in dom tt by FINSEQ_3:27;
then ( (apply (v1 ^ v2),t) . i = tt . i & tt . i = (apply v1,t) . i ) by A2, A3, FINSEQ_1:def 7;
hence (apply (v1 ^ v2),t) . i = (apply v1,t) . i ; :: thesis: verum