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 holds Start-At(0,S) c= p implies IC p = 0
proof let p be PartState of S;
A1: IC Start-At(0,S) = 0 by Def11;
 IC S in dom Start-At(0,S) by Def11;
  hence thesis by A1,GRFUNC_1:2;
end;
