reserve x, y, z, X for set,
  m, n for Nat,
  O for Ordinal,
  R, S for Relation;
reserve
  N for with_zero set,
  S for
  standard IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  L, l1 for Nat,
  J for Instruction of S,
  F for Subset of NAT;

theorem
  F = {n} implies LocSeq(F,S) = 0 .--> n
proof
  assume
A1: F = {n};
  then
A2: card F = {0} by CARD_1:30,49;
  {n} c= omega
  by ORDINAL1:def 12;
  then
A3: canonical_isomorphism_of(RelIncl order_type_of RelIncl {n}, RelIncl { n}
  ).0 = (0 .--> n).0 by CARD_5:38
    .= n by FUNCOP_1:72;
A4: dom LocSeq(F,S) = card F by Def1;
A5: for a being object st a in dom LocSeq(F,S) holds (LocSeq(F,S)).a
 = (0 .--> n ) . a
  proof
    let a be object;
    assume
A6: a in dom LocSeq(F,S);
    then
A7: a = 0 by A4,A2,TARSKI:def 1;
    thus (LocSeq(F,S)).a = (canonical_isomorphism_of
     (RelIncl order_type_of
    RelIncl F, RelIncl F).a) by A4,A6,Def1
      .= (0 .--> n).a by A1,A3,A7,FUNCOP_1:72;
  end;
  thus thesis by A1,A4,A5,CARD_1:30,49;
end;
