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 being 0-started PartState of S
  holds Initialize p = p
 proof let p be 0-started PartState of S;
   IC S in dom p & IC p = 0 by Def11;
  hence Initialize p = p by FUNCT_4:85,98;
 end;
