let S be non empty non void ManySortedSign ; :: thesis: for X being non-empty ManySortedSet of the carrier of S
for o being OperSymbol of S
for p being DTree-yielding FinSequence st [o, the carrier of S] -tree p in the Sorts of (FreeMSA X) . (the_result_sort_of o) holds
for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i)
let X be non-empty ManySortedSet of the carrier of S; :: thesis: for o being OperSymbol of S
for p being DTree-yielding FinSequence st [o, the carrier of S] -tree p in the Sorts of (FreeMSA X) . (the_result_sort_of o) holds
for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i)
let o be OperSymbol of S; :: thesis: for p being DTree-yielding FinSequence st [o, the carrier of S] -tree p in the Sorts of (FreeMSA X) . (the_result_sort_of o) holds
for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i)
let p be DTree-yielding FinSequence; :: thesis: ( [o, the carrier of S] -tree p in the Sorts of (FreeMSA X) . (the_result_sort_of o) implies for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i) )
A1: FreeMSA X = MSAlgebra(# (FreeSort X),(FreeOper X) #) by MSAFREE:def 14;
assume A2: [o, the carrier of S] -tree p in the Sorts of (FreeMSA X) . (the_result_sort_of o) ; :: thesis: for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i)

now :: thesis: for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i)

the carrier of S in { the carrier of S} by TARSKI:def 1;
then [o, the carrier of S] in [: the carrier' of S,{ the carrier of S}:] by ZFMISC_1:87;
then reconsider nt = [o, the carrier of S] as NonTerminal of (DTConMSA X) by MSAFREE:6;
set rso = the_result_sort_of o;
reconsider ao = the_arity_of o as Element of the carrier of S * ;
let i be Nat; :: thesis: ( i in dom (the_arity_of o) implies p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i) )
assume A3: i in dom (the_arity_of o) ; :: thesis: p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i)
then ao . i in rng ao by FUNCT_1:def 3;
then reconsider s = ao . i as SortSymbol of S ;
A4: the Sorts of (FreeMSA X) . s = FreeSort (X,s) by A1, MSAFREE:def 11
.= FreeSort (X,((the_arity_of o) /. i)) by A3, PARTFUN1:def 6 ;
[o, the carrier of S] -tree p in FreeSort (X,(the_result_sort_of o)) by A2, A1, MSAFREE:def 11;
then [o, the carrier of S] -tree p in { a where a is Element of TS (DTConMSA X) : ( ex x being set st
( x in X . (the_result_sort_of o) & a = root-tree [x,(the_result_sort_of o)] ) or ex o being OperSymbol of S st
( [o, the carrier of S] = a . {} & the_result_sort_of o = the_result_sort_of o ) ) } by MSAFREE:def 10;
then consider a being Element of TS (DTConMSA X) such that
A5: a = [o, the carrier of S] -tree p and
( ex x being set st
( x in X . (the_result_sort_of o) & a = root-tree [x,(the_result_sort_of o)] ) or ex o being OperSymbol of S st
( [o, the carrier of S] = a . {} & the_result_sort_of o = the_result_sort_of o ) ) ;
a . {} = [o, the carrier of S] by A5, TREES_4:def 4;
then consider ts being FinSequence of TS (DTConMSA X) such that
A6: a = nt -tree ts and
A7: nt ==> roots ts by DTCONSTR:10;
nt = Sym (o,X) by MSAFREE:def 9;
then A8: ts in (((FreeSort X) #) * the Arity of S) . o by A7, MSAFREE:10;
( dom p = Seg (len p) & dom (the_arity_of o) = Seg (len (the_arity_of o)) ) by FINSEQ_1:def 3;
then A9: i in dom p by A2, A3, Th10;
reconsider ts = ts as DTree-yielding FinSequence ;
ts = p by A5, A6, TREES_4:15;
hence p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i) by A9, A8, A4, MSAFREE:9; :: thesis: verum

end;

hence for i being Nat st i in dom (the_arity_of o) holds
p . i in the Sorts of (FreeMSA X) . ((the_arity_of o) . i) ; :: thesis: verum