reserve k for Nat;
reserve p for Prime;

theorem Ttool53a:
  p < 53 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
  proof
    assume p < 53;
    then 1+1 < p+1 & p < 52+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 47;
      hence thesis by Ttool47a;
    end;
    suppose 47 <= p <= 47+1 or 48 <= p <= 48+1 or 49 <= p <= 49+1 or 
      50 <= p <= 50+1 or 51 <= p <= 51+1;
      then p = 47 by XPRIMES0:48,49,50,51,52,NAT_1:9;
      hence thesis;
    end;
  end;
