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 FinSequence of S-Terms V st (len
a = len the_arity_of o or dom a = dom the_arity_of o) & ((for i being Nat st i
in dom a for t being Term of S,V st t = a.i holds the_sort_of t = (the_arity_of
o).i) or for i being Nat st i in dom a for t being Term of S,V st t = a.i holds
  the_sort_of t = (the_arity_of o)/.i) holds a is ArgumentSeq of Sym(o,V)
proof
  let o be OperSymbol of S, a be FinSequence of S-Terms V such that
A1: len a = len the_arity_of o or dom a = dom the_arity_of o and
A2: (for i being Nat st i in dom a for t being Term of S,V st t = a.i
holds the_sort_of t = (the_arity_of o).i) or for i being Nat st i in dom a for
  t being Term of S,V st t = a.i holds the_sort_of t = (the_arity_of o)/.i;
A3: now
    let i be Nat;
    assume i in dom a;
    then
A4: a.i in rng a by FUNCT_1:def 3;
    rng a c= S-Terms V by FINSEQ_1:def 4;
    hence a.i in S-Terms V by A4;
  end;
  now
    per cases by A2;
    case
A5:   for i being Nat st i in dom a for t being Term of S,V st t = a.i
      holds the_sort_of t = (the_arity_of o).i;
      let i be Nat;
      assume
A6:   i in dom a;
      then reconsider t = a.i as Term of S,V by A3;
      take t;
      thus t = a.i & the_sort_of t = (the_arity_of o).i by A5,A6;
    end;
    case
A7:   for i being Nat st i in dom a for t being Term of S,V st t = a.
      i holds the_sort_of t = (the_arity_of o)/.i;
      let i be Nat;
      assume
A8:   i in dom a;
      then reconsider t = a.i as Term of S,V by A3;
      take t;
      thus t = a.i & the_sort_of t = (the_arity_of o)/.i by A7,A8;
    end;
  end;
  hence thesis by A1,Th24;
end;
