reserve x,y for set,
  n for Nat;

theorem Th8:
  for f be with_zero non empty FinSequence of NAT,D be
  disjoint_with_NAT set holds FreeUnivAlgZAO(f,D) is with_const_op
proof
  let f be with_zero non empty FinSequence of NAT, D be disjoint_with_NAT set;
  set A = DTConUA(f,D), AA = FreeUnivAlgZAO(f,D);
A1: dom f = Seg len f by FINSEQ_1:def 3;
  0 in rng f by Def2;
  then consider n being Nat such that
A2: n in dom f and
A3: f.n = 0 by FINSEQ_2:10;
A4: len FreeOpSeqZAO(f,D) = len f & dom FreeOpSeqZAO(f,D) = Seg len
  FreeOpSeqZAO (f,D) by Def17,FINSEQ_1:def 3;
  then (the charact of AA).n = FreeOpZAO(n,f,D) by A2,A1,Def17;
  then reconsider o = FreeOpZAO(n,f,D) as operation of AA by A2,A4,A1,
FUNCT_1:def 3;
  take o;
A5: dom o = (arity o)-tuples_on (the carrier of AA) by MARGREL1:22;
  f/.n = f.n & dom(FreeOpZAO(n,f,D)) = (f/.n)-tuples_on (TS A) by A2,Def16,
PARTFUN1:def 6;
  hence thesis by A3,A5,FINSEQ_2:110;
end;
