let S be non empty non void 1-1-connectives bool-correct 4,1 integer 11,1,1 -array 11 array-correct BoolSignature ; for X being V3() ManySortedSet of the carrier of S
for T being non-empty b1,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer-array VarMSAlgebra over S
for C being bool-correct 4,1 integer 11,1,1 -array image of T
for I being integer SortSymbol of S
for u being ManySortedFunction of FreeGen T, the Sorts of C
for t being Element of T,(the_array_sort_of S)
for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let X be V3() ManySortedSet of the carrier of S; for T being non-empty X,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer-array VarMSAlgebra over S
for C being bool-correct 4,1 integer 11,1,1 -array image of T
for I being integer SortSymbol of S
for u being ManySortedFunction of FreeGen T, the Sorts of C
for t being Element of T,(the_array_sort_of S)
for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let T be non-empty X,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer-array VarMSAlgebra over S; for C being bool-correct 4,1 integer 11,1,1 -array image of T
for I being integer SortSymbol of S
for u being ManySortedFunction of FreeGen T, the Sorts of C
for t being Element of T,(the_array_sort_of S)
for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let C be bool-correct 4,1 integer 11,1,1 -array image of T; for I being integer SortSymbol of S
for u being ManySortedFunction of FreeGen T, the Sorts of C
for t being Element of T,(the_array_sort_of S)
for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let I be integer SortSymbol of S; for u being ManySortedFunction of FreeGen T, the Sorts of C
for t being Element of T,(the_array_sort_of S)
for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let u be ManySortedFunction of FreeGen T, the Sorts of C; for t being Element of T,(the_array_sort_of S)
for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let t be Element of T,(the_array_sort_of S); for t1, t2 being Element of T,I holds ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
let t1, t2 be Element of T,I; ((t,t1) <- t2) value_at (C,u) = ((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
set o = In (( the connectives of S . 12), the carrier' of S);
consider f being ManySortedFunction of T,C such that
A1:
( f is_homomorphism T,C & u = f || (FreeGen T) )
by MSAFREE4:46;
A2:
t2 value_at (C,u) = (f . I) . t2
by A1, Th28;
A3:
t value_at (C,u) = (f . (the_array_sort_of S)) . t
by A1, Th28;
A4:
((t,t1) <- t2) value_at (C,u) = (f . (the_array_sort_of S)) . ((t,t1) <- t2)
by A1, Th28;
A5:
( the_arity_of (In (( the connectives of S . 12), the carrier' of S)) = <*(the_array_sort_of S),I,I*> & the_result_sort_of (In (( the connectives of S . 12), the carrier' of S)) = the_array_sort_of S )
by Th76;
then
Args ((In (( the connectives of S . 12), the carrier' of S)),T) = product <*( the Sorts of T . (the_array_sort_of S)),( the Sorts of T . I),( the Sorts of T . I)*>
by Th24;
then reconsider p = <*t,t1,t2*> as Element of Args ((In (( the connectives of S . 12), the carrier' of S)),T) by FINSEQ_3:125;
thus ((t,t1) <- t2) value_at (C,u) =
(Den ((In (( the connectives of S . 12), the carrier' of S)),C)) . (f # p)
by A1, A4, A5
.=
(Den ((In (( the connectives of S . 12), the carrier' of S)),C)) . <*((f . (the_array_sort_of S)) . t),((f . I) . t1),((f . I) . t2)*>
by A5, Th27
.=
((t value_at (C,u)),(t1 value_at (C,u))) <- (t2 value_at (C,u))
by A1, A2, A3, Th28
; verum