reserve k for Nat;
reserve p for Prime;

theorem Ttool89a:
  p < 89 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or 
  p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or 
  p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or 
  p = 79 or p = 83
  proof
    assume p < 89;
    then 1+1 < p+1 & p < 88+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 83;
      hence thesis by Ttool83a;
    end;
    suppose 83 <= p <= 83+1 or 84 <= p <= 84+1 or 85 <= p <= 85+1 or 
      86 <= p <= 86+1 or 87 <= p <= 87+1;
      then p = 83 by XPRIMES0:84,85,86,87,88,NAT_1:9;
      hence thesis;
    end;
  end;
