
theorem Th17:
  for X being disjoint_with_NAT non empty set
  for S being non empty FinSequence of NAT for i being Nat st i in dom S
  for p being FinSequence of FreeUnivAlgNSG(S,X) st len p = S.i holds
  Den(In(i, dom the charact of FreeUnivAlgNSG(S,X)),FreeUnivAlgNSG(S,X)).p
  = i-tree p
proof
  let X be disjoint_with_NAT non empty set;
  let S be non empty FinSequence of NAT;
  reconsider S9 = S as non empty FinSequence of omega;
  set G = DTConUA(S,X);
  set A = FreeUnivAlgNSG(S,X);
  let i be Nat;
  assume
A1: i in dom S;
  then
A2: S9/.i = S.i by PARTFUN1:def 6;
  let p be FinSequence of A;
  assume len p = S.i;
  then p is Element of (S9/.i)-tuples_on TS G by A2,FINSEQ_2:92;
  then p in (S9/.i)-tuples_on TS G;
  then
A3: p in dom FreeOpNSG(i,S,X) by A1,FREEALG:def 10;
  len the charact of A = len S by FREEALG:def 11;
  then dom the charact of A = dom S by FINSEQ_3:29;
  then In(i, dom the charact of A) = i by A1,SUBSET_1:def 8;
  hence Den(In(i, dom the charact of A), A).p
  = FreeOpNSG(i,S,X).p by FREEALG:def 11
    .= Sym(i,S,X)-tree p by A1,A3,FREEALG:def 10
    .= i-tree p by A1,FREEALG:def 9;
end;
