reserve I for non empty set,
  J for ManySortedSet of I,
  S for non void non empty ManySortedSign,
  i for Element of I,
  c for set,
  A for MSAlgebra-Family of I,S,
  EqR for Equivalence_Relation of I,
  U0,U1,U2 for MSAlgebra over S,
  s for SortSymbol of S,
  o for OperSymbol of S,
  f for Function;

theorem Th11:
  for e be Element of Args(o,U1) st e = {} & the_arity_of o = {} &
Args(o,U1) <> {} & Args(o,U2) <> {} for F be ManySortedFunction of U1,U2 holds
  F#e = {}
proof
  let e be Element of Args(o,U1) such that
A1: e = {} and
A2: the_arity_of o = {} and
A3: Args(o,U1) <> {} & Args(o,U2) <> {};
  reconsider e1 = e as Function by A1;
  let F be ManySortedFunction of U1,U2;
A4: dom (F*the_arity_of o) = {} by A2;
  then rng (F*the_arity_of o) = {} by RELAT_1:42;
  then (F*the_arity_of o) is Function of {},{} by A4,FUNCT_2:1;
  then
A5: e1 in product (doms (F*the_arity_of o))
  by A1,CARD_3:10,FUNCT_6:23,TARSKI:def 1;
A6: F#e = (Frege(F*the_arity_of o)).e by A3,MSUALG_3:def 5
    .= (F*the_arity_of o)..e1 by A5,PRALG_2:def 2;
  then reconsider fn = F#e as Function;
  dom fn = {} /\ dom e1 by A4,A6,PRALG_1:def 19;
  hence thesis;
end;
