
theorem
  for S being non void non empty ManySortedSign, A being MSAlgebra over
S, o being OperSymbol of S, 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)
proof
  let S be non void non empty ManySortedSign, A be MSAlgebra over S, o be
  OperSymbol of S, p be FinSequence such 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);
  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 D by A3,RELAT_1:27;
A5: now
    let x be object;
    assume
A6: x in dom D;
    then reconsider k = x as Nat;
    D.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 D.x by A2,A4,A6;
  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 thesis by A4,A5,CARD_3:def 5;
end;
