reserve x,N for set,
        k for Nat;
reserve N for with_zero set;

theorem Th2:
  for S being non empty with_non-empty_values Mem-Struct over N
  for s being State of S holds IC S in dom s
proof
  let S be non empty with_non-empty_values Mem-Struct over N;
  let s be State of S;
  dom s = the carrier of S by PARTFUN1:def 2;
  hence thesis;
end;
