let S be non empty non void ManySortedSign ; :: thesis: for A being MSAlgebra over S
for o being OperSymbol of S
for p being FinSequence st len p = len (the_arity_of o) & ( for k being Nat st k in dom p holds
p . k in the Sorts of A . ((the_arity_of o) /. k) ) holds
p in Args (o,A)
let A be MSAlgebra over S; :: thesis: for o being OperSymbol of S
for p being FinSequence st len p = len (the_arity_of o) & ( for k being Nat st k in dom p holds
p . k in the Sorts of A . ((the_arity_of o) /. k) ) holds
p in Args (o,A)
let o be OperSymbol of S; :: thesis: for p being FinSequence st len p = len (the_arity_of o) & ( for k being Nat st k in dom p holds
p . k in the Sorts of A . ((the_arity_of o) /. k) ) holds
p in Args (o,A)
let p be FinSequence; :: thesis: ( len p = len (the_arity_of o) & ( for k being Nat st k in dom p holds
p . k in the Sorts of A . ((the_arity_of o) /. k) ) implies p in Args (o,A) )
assume that
A1: len p = len (the_arity_of o) and
A2: for k being Nat st k in dom p holds
p . k in the Sorts of A . ((the_arity_of o) /. k) ; :: thesis: p in Args (o,A)
set D = the Sorts of A * (the_arity_of o);
A3: dom p = dom (the_arity_of o) by A1, FINSEQ_3:29;
rng (the_arity_of o) c= the carrier of S ;
then rng (the_arity_of o) c= dom the Sorts of A by PARTFUN1:def 2;
then A4: dom p = dom ( the Sorts of A * (the_arity_of o)) by A3, RELAT_1:27;

A5: now :: thesis: for x being object st x in dom ( the Sorts of A * (the_arity_of o)) holds
p . x in ( the Sorts of A * (the_arity_of o)) . x

let x be object ; :: thesis: ( x in dom ( the Sorts of A * (the_arity_of o)) implies p . x in ( the Sorts of A * (the_arity_of o)) . x )
assume A6: x in dom ( the Sorts of A * (the_arity_of o)) ; :: thesis: p . x in ( the Sorts of A * (the_arity_of o)) . x
then reconsider k = x as Nat ;
( the Sorts of A * (the_arity_of o)) . k = the Sorts of A . ((the_arity_of o) . k) by A6, FUNCT_1:12
.= the Sorts of A . ((the_arity_of o) /. k) by A3, A4, A6, PARTFUN1:def 6 ;
hence p . x in ( the Sorts of A * (the_arity_of o)) . x by A2, A4, A6; :: thesis: verum

end;

dom the Arity of S = the carrier' of S by FUNCT_2:def 1;
then (( the Sorts of A #) * the Arity of S) . o = ( the Sorts of A #) . (the_arity_of o) by FUNCT_1:13
.= product ( the Sorts of A * (the_arity_of o)) by FINSEQ_2:def 5 ;
hence p in Args (o,A) by A4, A5, CARD_3:def 5; :: thesis: verum