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