reserve k for Nat;
reserve p for Prime;

theorem Ttool163a:
  p < 163 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 or p = 97 or p = 101 or p = 103 or p = 107 or 
  p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or 
  p = 151 or p = 157
  proof
    assume p < 163;
    then 1+1 < p+1 & p < 162+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 157;
      hence thesis by Ttool157a;
    end;
    suppose 157 <= p <= 157+1 or 158 <= p <= 158+1 or 159 <= p <= 159+1 or 
      160 <= p <= 160+1 or 161 <= p <= 161+1;
      then p = 157 by XPRIMES0:158,159,160,161,162,NAT_1:9;
      hence thesis;
    end;
  end;
