reserve G for Graph,
  k, m, n for Nat;
reserve G for non void Graph;

theorem Th23:
  for S being non void non empty ManySortedSign, A being non-empty
MSAlgebra over S, o be OperSymbol of S st (the Arity of S).o = {} holds dom Den
  (o, A) = {{}}
proof
  dom {} = {} & rng {} = {};
  then reconsider b = {} as Function of {},{} by FUNCT_2:1;
  let S be non void non empty ManySortedSign, A be non-empty MSAlgebra over S,
  o be OperSymbol of S such that
A1: (the Arity of S).o = {};
A2: dom (the Arity of S) = the carrier' of S by FUNCT_2:def 1;
  then dom ((the Sorts of A)# qua ManySortedSet of(the carrier of S)*) = (the
  carrier of S)* & (the Arity of S).o in rng (the Arity of S) by FUNCT_1:def 3
,PARTFUN1:def 2;
  then
A3: o in dom ((the Sorts of A)# * the Arity of S) by A2,FUNCT_1:11;
  thus dom Den(o,A) = Args(o,A) by FUNCT_2:def 1
    .= ((the Sorts of A)# * the Arity of S).o by MSUALG_1:def 4
    .= (the Sorts of A)# . ((the Arity of S).o) by A3,FUNCT_1:12
    .= (the Sorts of A)# . (the_arity_of o) by MSUALG_1:def 1
    .= product ((the Sorts of A) * (the_arity_of o) qua Function) by
FINSEQ_2:def 5
    .= product ((the Sorts of A) * b) by A1,MSUALG_1:def 1
    .= {{}} by CARD_3:10;
end;
