reserve S for non empty non void ManySortedSign,
  A for MSAlgebra over S;

theorem Th7:
 for S being non empty non void ManySortedSign, o being OperSymbol
  of S for A being MSAlgebra over S, a being Function st a in Args(o,A)
  for i being Nat, x being Element of A,(the_arity_of o)/.i holds a+*(i,x)
  in Args(o,A)
proof
  let S be non empty non void ManySortedSign, o be OperSymbol of S;
  let A be MSAlgebra over S;
A1: dom ((the Sorts of A)*the_arity_of o) = dom the_arity_of o by PRALG_2:3;
  let a be Function such that
A2: a in Args(o,A);
A3: dom a = dom the_arity_of o by A2,Th2;
  let i be Nat;
  let x be Element of A,(the_arity_of o)/.i;
A4: Args(o,A) = product ((the Sorts of A)*the_arity_of o) by PRALG_2:3;
A5: now
    let j be object;
    assume
A6: j in dom a;
    then reconsider k = j as Element of NAT by A3;
A7: ((the Sorts of A)*the_arity_of o).k = (the Sorts of A).((the_arity_of
    o) .k) by A3,A6,FUNCT_1:13;
A8: (the_arity_of o)/.k = (the_arity_of o).k by A3,A6,PARTFUN1:def 6;
    then
A9: ((the Sorts of A)*the_arity_of o).j <> {} by A2,A3,A6,A7,Th3;
    per cases;
    suppose
A10:  j = i;
      then (a+*(i,x)).j = x by A6,FUNCT_7:31;
      hence (a+*(i,x)).j in ((the Sorts of A)*the_arity_of o).j by A8,A7,A9,A10
;
    end;
    suppose
      j <> i;
      then (a+*(i,x)).j = a.j by FUNCT_7:32;
      hence
      (a+*(i,x)).j in ((the Sorts of A)*the_arity_of o).j by A2,A4,A1,A3,A6,
CARD_3:9;
    end;
  end;
  dom (a+*(i,x)) = dom a by FUNCT_7:30;
  hence thesis by A4,A1,A3,A5,CARD_3:9;
end;
