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 S being IC-Ins-separated non empty with_non-empty_values Mem-Struct over N
 for p being PartState of S, k being Nat
 holds DataPart p c= IncIC(p,k)
proof
 let S be IC-Ins-separated non empty with_non-empty_values Mem-Struct over N;
 let p being PartState of S, k be Nat;
  DataPart IncIC(p,k) = DataPart p by Th51;
 hence DataPart p c= IncIC(p,k) by RELAT_1:59;
end;
