reserve k for Nat;
reserve p for Prime;

theorem Ttool41a:
  p < 41 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
  proof
    assume p < 41;
    then 1+1 < p+1 & p < 40+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 37;
      hence thesis by Ttool37a;
    end;
    suppose 37 <= p <= 37+1 or 38 <= p <= 38+1 or 39 <= p <= 39+1;
      then p = 37 by XPRIMES0:38,39,40,NAT_1:9;
      hence thesis;
    end;
  end;
