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 p,q being PartState of S holds IC S in dom(p +* Initialize q)
proof let p,q be PartState of S;
A1: dom(p +* Initialize q) =  dom p \/ dom Initialize q by FUNCT_4:def 1;
  IC S in dom Initialize q by Th48;
 hence IC S in dom(p +* Initialize q) by A1,XBOOLE_0:def 3;
end;
