reserve S for non void non empty ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S;
reserve A for MSAlgebra over S,
  t for Term of S,V;

theorem
  for o being OperSymbol of S, a being ArgumentSeq of Sym(o,V) for i
  being Nat st i in dom a for t being Term of S,V st t = a.i holds t = (a qua
FinSequence of S-Terms V qua non empty set)/.i & the_sort_of t = (the_arity_of
  o).i & the_sort_of t = (the_arity_of o)/.i
proof
  let o be OperSymbol of S, a be ArgumentSeq of Sym(o,V);
  let i be Nat;
  assume i in dom a;
  then
  ex t being Term of S,V st t = a.i & t = (a qua FinSequence of S-Terms V
qua non empty set)/.i & the_sort_of t = (the_arity_of o).i & the_sort_of t = (
  the_arity_of o)/.i by Lm8;
  hence thesis;
end;
