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
  p +* Start-At(k,S) c= s implies IC s = k
proof
  assume
A1: p +* Start-At(k,S) c= s;
  IC S in dom (p +* Start-At( k,S)) by Th27;
  hence IC s = IC (p +* Start-At( k,S)) by A1,GRFUNC_1:2
    .=  k by Th16;
end;
