reserve k for Nat;
reserve p for Prime;

theorem Ttool97a:
  p < 97 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 or p = 89
  proof
    assume p < 97;
    then 1+1 < p+1 & p < 96+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 89;
      hence thesis by Ttool89a;
    end;
    suppose 89 <= p <= 89+1 or 90 <= p <= 90+1 or 91 <= p <= 91+1 or 
      92 <= p <= 92+1 or 93 <= p <= 93+1 or 94 <= p <= 94+1 or 
      95 <= p <= 95+1;
      then p = 89 by XPRIMES0:90,91,92,93,94,95,96,NAT_1:9;
      hence thesis;
    end;
  end;
