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
 dom  p c= {IC S} \/ dom DataPart p
proof
 set S0 = Start-At(0,S);
 per cases;
 suppose IC S in dom p;
  hence thesis by Th24;
 end;
 suppose
A1: not IC S in dom p;
A3:  dom (p +* S0)
     = {IC S} \/ dom DataPart(p +* S0) by Th24,Th27
    .= {IC S} \/ dom(DataPart p +* DataPart S0) by FUNCT_4:71
    .= {IC S} \/ dom(DataPart p +* {}) by Th20
    .= {IC S} \/ dom DataPart p;
  now assume dom p meets dom S0;
   then consider x being object such that
A4: x in dom p and
A5: x in dom S0 by XBOOLE_0:3;
   thus contradiction by A4,A1,A5,TARSKI:def 1;
  end;
  then p c= p +* S0 by FUNCT_4:32;
 hence dom  p c= {IC S} \/ dom DataPart p by A3,RELAT_1:11;
 end;
end;
