reserve x,N for set,
        k for Nat;
reserve N for with_zero set;
reserve S for IC-Ins-separated non empty with_non-empty_values
     Mem-Struct over N;
reserve s for State of S;
reserve p for PartState of S;

theorem
  for S being with_non-empty_values Mem-Struct over N,
      p being Element of FinPartSt S holds
  p is FinPartState of S
proof
  let S be with_non-empty_values Mem-Struct over N;
  let p be Element of FinPartSt S;
  p in FinPartSt S;
  then ex q being Element of sproduct the_Values_of S st q = p & q is
  finite;
  hence thesis;
end;
