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 Th24:
  for o being OperSymbol of S, a being FinSequence 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
ex t being Term of S,V st t = a.i & the_sort_of t = (the_arity_of o).i) or for
i being Nat st i in dom a ex t being Term of S,V st t = a.i & the_sort_of t = (
  the_arity_of o)/.i) holds a is ArgumentSeq of Sym(o,V)
proof
  set X = V;
  let o be OperSymbol of S, a be FinSequence 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 ex t being Term of S,V st t = a.i &
the_sort_of t = (the_arity_of o).i) or for i being Nat st i in dom a ex t being
  Term of S,V st t = a.i & the_sort_of t = (the_arity_of o)/.i;
  rng a c= TS DTConMSA X
  proof
    let x be object;
    assume x in rng a;
    then consider i being object such that
A3: i in dom a and
A4: x = a.i by FUNCT_1:def 3;
    reconsider i as Nat by A3;
    (ex t being Term of S,V st t = a.i & the_sort_of t = (the_arity_of o).
i) or ex t being Term of S,V st t = a.i & the_sort_of t = (the_arity_of o)/.i
    by A2,A3;
    hence thesis by A4;
  end;
  then reconsider p = a as FinSequence of TS DTConMSA X by FINSEQ_1:def 4;
A5: dom a = dom the_arity_of o by A1,FINSEQ_3:29;
  now
    let n be Nat;
    assume
A6: n in dom p;
    thus p.n in FreeSort(X,(the_arity_of o)/.n)
    proof
      per cases by A2,A6;
      suppose
        ex t being Term of S,V st t = a.n & the_sort_of t = (
        the_arity_of o).n;
        then consider t be Term of S,V such that
A7:     t = a.n and
A8:     the_sort_of t = (the_arity_of o).n;
        the_sort_of t = (the_arity_of o)/.n by A5,A6,A8,PARTFUN1:def 6;
        hence thesis by A7,Def5;
      end;
      suppose
        ex t being Term of S,V st t = a.n & the_sort_of t = (
        the_arity_of o)/.n;
        hence thesis by Def5;
      end;
    end;
  end;
  then p in ((FreeSort X)# * (the Arity of S)).o by A5,MSAFREE:9;
  then
A9: Sym(o, X) ==> roots p by MSAFREE:10;
  S-Terms V = TS DTConMSA X;
  hence thesis by A9,Th21;
end;
