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 A being non empty with_non-empty_values Mem-Struct over N, s
  being State of A, o being Object of A holds s.o in Values o
proof
  let A be non empty with_non-empty_values Mem-Struct over N,
      s be State of A, o be Object of A;
  dom s = the carrier of A by PARTFUN1:def 2;
  hence thesis by FUNCT_1:def 14;
end;
