reserve n,k,b for Nat, i for Integer;

theorem Th41:
  11 divides n iff 11 divides
  value(mid(digits(n,10),2,len(digits(n,10))),10) - digits(n,10).0
  proof
    reconsider p=11 as Prime by NAT_4:27;
    A1: 10*1 + 1 = p*1;
    p,10 are_coprime by EULER_1:2; then
    11 divides n iff 11 divides
    value(mid(digits(n,10),2,len(digits(n,10))),10) - 1*digits(n,10).0
    by Th37,A1;
    hence thesis;
  end;
