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 Th24:
 for S being IC-Ins-separated non empty with_non-empty_values Mem-Struct over N
 for p being PartState of S st IC S in dom p
  holds dom  p = {IC S} \/ dom DataPart p
proof
 let S be IC-Ins-separated non empty with_non-empty_values Mem-Struct over N;
 let p be PartState of S;
 assume IC S in dom p;
  then
A1: p = Start-At(IC p,S) +* DataPart p by Th19;
  dom p =  dom Start-At(IC p,S) \/ dom DataPart p by A1,FUNCT_4:def 1
    .= {IC S} \/ dom DataPart p;
 hence thesis;
end;
