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

theorem Th80:
  89 divides n iff 89 divides
  value(mid(digits(n,10),2,len(digits(n,10))),10) + 9*digits(n,10).0
  proof
    reconsider p=89 as Prime by Th34;
    A1: 10*9 - 1 = p*1;
    p,10 are_coprime by EULER_1:2;
    hence thesis by Th38,A1;
  end;
