let S be non empty non void ManySortedSign ; :: thesis:
let X be V5() ManySortedSet of S; :: thesis:
let w be ManySortedFunction of X, the carrier of S --> NAT; :: thesis:
deffunc H₂( set , set ) -> set = root-tree [((w . $2) . $1),$2];
deffunc H₃( OperSymbol of S, FinSequence) -> set = (Sym ($1,( the carrier of S --> NAT))) -tree $2;
let F be ManySortedSet of S -Terms X; :: thesis:
assume that
A1: for s being SortSymbol of S
for x being Element of X . s holds F . (root-tree [x,s]) = H₂(x,s) and
A2: for o being OperSymbol of S
for p being ArgumentSeq of Sym (o,X) holds F . ((Sym (o,X)) -tree p) = H₃(o,F * p) ; :: thesis:
let o be OperSymbol of S; :: thesis:
let p be ArgumentSeq of Sym (o,X); :: thesis:
rng p c= dom F

proof

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in rng p ; :: thesis:
then consider x being object such that
A3: ( x in dom p & y = p . x ) by FUNCT_1:def 3;
reconsider x = x as Nat by A3;
y is Term of S,X by A3, MSATERM:22;
then y in S -Terms X ;
hence y in dom F by PARTFUN1:def 2; :: thesis:

end;

then A4: dom (F * p) = dom p by RELAT_1:27;
A5: dom p = dom (the_arity_of o) by MSATERM:22;
A6: S -Terms X = Union the Sorts of (FreeMSA X) by MSATERM:13
.= Union the Sorts of (Free (S,X)) by MSAFREE3:31 ;
X is_transformable_to the carrier of S --> NAT by Th21;
then rngs w is ManySortedSubset of the carrier of S --> NAT by PBOOLE:def 18, MSSUBFAM:17;
then A7: Free (S,(rngs w)) is MSSubAlgebra of Free (S,( the carrier of S --> NAT)) by Th59;

now :: thesis:

let i be Nat; :: thesis:
assume A8: i in dom p ; :: thesis:
then reconsider t = p . i as Term of S,X by MSATERM:22;
F . t = # (t,w) by A1, A2, A6, Th55;
then reconsider tt = F . t as Element of the Sorts of (Free (S,(rngs w))) . (the_sort_of t) by Th60;
A9: the Sorts of (Free (S,(rngs w))) is MSSubset of (Free (S,( the carrier of S --> NAT))) by A7, MSUALG_2:def 9;
reconsider tt = tt as Term of S,(rngs w) by Th42;
take tt = tt; :: thesis:
thus tt = (F * p) . i by A8, FUNCT_1:13; :: thesis:
( the Sorts of (Free (S,(rngs w))) . (the_sort_of t) c= the Sorts of (Free (S,( the carrier of S --> NAT))) . (the_sort_of t) & tt in the Sorts of (Free (S,(rngs w))) . (the_sort_of t) ) by A9, PBOOLE:def 2, PBOOLE:def 18;
then tt in the Sorts of (FreeMSA (rngs w)) . (the_sort_of t) by MSAFREE3:31;
then tt in FreeSort ((rngs w),(the_sort_of t)) by MSAFREE:def 11;
hence the_sort_of tt = the_sort_of t by MSATERM:def 5
.= (the_arity_of o) . i by A8, MSATERM:23 ;
:: thesis:

end;

then F * p is ArgumentSeq of Sym (o,(rngs w)) by A4, A5, MSATERM:24;
then F * p is Element of Args (o,(FreeMSA (rngs w))) by INSTALG1:1;
then F * p is Element of Args (o,(Free (S,(rngs w)))) by MSAFREE3:31;
then ( F * p in Args (o,(Free (S,(rngs w)))) & Args (o,(Free (S,(rngs w)))) c= Args (o,(Free (S,( the carrier of S --> NAT)))) ) by A7, MSAFREE3:37;
hence ( F * p in Args (o,(Free (S,(rngs w)))) & F * p in Args (o,(Free (S,( the carrier of S --> NAT)))) ) ; :: thesis: