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 PartState of S st Start-At(k,S) c= p holds IC p = k
proof
 let p be PartState of S;
 assume Start-At(k,S) c= p;
  then p is k-started by Th29;
 hence IC p = k;
end;
