reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;
reserve m,n for Nat,
  s,w for FinSequence of NAT;
reserve i,j,e,u for set,
        n for Nat;

theorem
  for a being set for M being ManySortedSet of {a} st M = a .--> D holds
  (M#* *-->a).n = Funcs(Seg n, D)
proof
  let a be set;
  let M be ManySortedSet of {a} such that
A1: M = a .--> D;
  a is Element of {a} by TARSKI:def 1;
  then n |-> a is FinSequence of {a} by Th61;
  then
A2: n |-> a in {a}* by FINSEQ_1:def 11;
   reconsider nn=n as Element of NAT by ORDINAL1:def 12;
  dom(*-->a) = NAT by FUNCT_2:def 1;
  hence (M#* *-->a).n = M#.((*-->a).nn) by FUNCT_1:13
    .= M#.(nn|->a) by Def6
    .= product((a .--> D)*(n|->a)) by A1,A2,Def5
    .= product(n |-> D) by Th142
    .= Funcs(Seg n, D) by CARD_3:11;
end;
