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 l1,l2,k being Nat st Start-At(l1,S) = Start-At(l2,S)
  holds Start-At(l1 -' k,S) = Start-At(l2 -' k,S)
proof
  let l1,l2,k be Nat;
  assume Start-At(l1,S) = Start-At(l2,S);
  then {[IC S, l1]} = Start-At(l2,S) by FUNCT_4:82
    .= {[IC S, l2]} by FUNCT_4:82;
  then [IC S, l1] = [IC S, l2] by ZFMISC_1:3;
  hence thesis by XTUPLE_0:1;
end;
