XPRIMET1

:: Tools for Computing Prime Numbers - Part I
::  by Library Committee
::
:: Received October 31, 2022
:: Copyright (c) 2022-2026 Association of Mizar Users
::           (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies NUMBERS, NAT_1, RELAT_1, XXREAL_0, ARYTM_3, NEWTON;
 notations ORDINAL1, XCMPLX_0, INT_2, XXREAL_0;
 constructors NEWTON, VALUED_0, MEMBERED;
 registrations XREAL_0, INT_1, MEMBERED;
 requirements REAL, NUMERALS, SUBSET, ARITHM;
 theorems XREAL_1, INT_2, XPRIMES0, XXREAL_0, NAT_4, NAT_1;

begin

reserve k for Nat;
reserve p for Prime;

theorem Th1:
  p < 3 implies p = 2
  proof
    assume p < 3;
    then p < 2+1;
    then
A1: p <= 2 by NAT_1:13;
    1 < p by INT_2:def 4;
    then 1+1 <= p by NAT_1:13;
    hence thesis by A1,XXREAL_0:1;
  end;

theorem 
  p*p <= k < 9 implies p = 2
  proof
    assume p*p <= k < 9;
    then p*p < 3*3 by XXREAL_0:2;
    hence thesis by Th1,NAT_4:1;
  end;

theorem Ttool5a:
  p < 5 implies p = 2 or p = 3
  proof
    assume p < 5;
    then 1+1 < p+1 & p < 4+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 3;
      hence thesis by Th1;
    end;
    suppose 3 <= p <= 3+1;
      hence thesis by XPRIMES0:4,NAT_1:9;
    end;
  end;

theorem
  p*p <= k < 25 implies p = 2 or p = 3
  proof
    assume p*p <= k < 25;
    then p*p < 5*5 by XXREAL_0:2;
    hence thesis by Ttool5a,NAT_4:1;
  end;

theorem Ttool7a:
  p < 7 implies p = 2 or p = 3 or p = 5
  proof
    assume p < 7;
    then 1+1 < p+1 & p < 6+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 5;
      hence thesis by Ttool5a;
    end;
    suppose 5 <= p <= 5+1;
      then p = 5 by XPRIMES0:6,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 49 implies p = 2 or p = 3 or p = 5
  proof
    assume p*p <= k < 49;
    then p*p < 7*7 by XXREAL_0:2;
    hence thesis by Ttool7a,NAT_4:1;
  end;

theorem Ttool11a:
  p < 11 implies p = 2 or p = 3 or p = 5 or p = 7
  proof
    assume p < 11;
    then 1+1 < p+1 & p < 10+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 7;
      hence thesis by Ttool7a;
    end;
    suppose 7 <= p <= 7+1 or 8 <= p <= 8+1 or 9 <= p <= 9+1;
      then p = 7 by XPRIMES0:10,8,9,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 121 implies p = 2 or p = 3 or p = 5 or p = 7
  proof
    assume p*p <= k < 121;
    then p*p < 11*11 by XXREAL_0:2;
    hence thesis by Ttool11a,NAT_4:1;
  end;

theorem Ttool13a:
  p < 13 implies p = 2 or p = 3 or p = 5 or p = 7 or p = 11
  proof
    assume p < 13;
    then 1+1 < p+1 & p < 12+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 11;
      hence thesis by Ttool11a;
    end;
    suppose 11 <= p <= 11+1;
      hence thesis by XPRIMES0:12,NAT_1:9;
    end;
  end;

theorem
  p*p <= k < 169 implies p = 2 or p = 3 or p = 5 or p = 7 or p = 11
  proof
    assume p*p <= k < 169;
    then p*p < 13*13 by XXREAL_0:2;
    hence thesis by Ttool13a,NAT_4:1;
  end;

theorem Ttool17a:
  p < 17 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13
  proof
    assume p < 17;
    then 1+1 < p+1 & p < 16+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 13;
      hence thesis by Ttool13a;
    end;
    suppose 13 <= p <= 13+1 or 14 <= p <= 14+1 or 15 <= p <= 15+1;
      then p = 13 by XPRIMES0:14,15,16,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 289 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13
  proof
    assume p*p <= k < 289;
    then p*p < 17*17 by XXREAL_0:2;
    hence thesis by Ttool17a,NAT_4:1;
  end;

theorem Ttool19a:
  p < 19 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17
  proof
    assume p < 19;
    then 1+1 < p+1 & p < 18+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 17;
      hence thesis by Ttool17a;
    end;
    suppose 17 <= p <= 17+1;
      hence thesis by XPRIMES0:18,NAT_1:9;
    end;
  end;

theorem
  p*p <= k < 361 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17
  proof
    assume p*p <= k < 361;
    then p*p < 19*19 by XXREAL_0:2;
    hence thesis by Ttool19a,NAT_4:1;
  end;

theorem Ttool23a:
  p < 23 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or 
  p = 19
  proof
    assume p < 23;
    then 1+1 < p+1 & p < 22+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 19;
      hence thesis by Ttool19a;
    end;
    suppose 19 <= p <= 19+1 or 20 <= p <= 20+1 or 21 <= p <= 21+1;
      then p = 19 by XPRIMES0:20,21,22,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 529 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or 
  p = 19
  proof
    assume p*p <= k < 529;
    then p*p < 23*23 by XXREAL_0:2;
    hence thesis by Ttool23a,NAT_4:1;
  end;

theorem Ttool29a:
  p < 29 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
  proof
    assume p < 29;
    then 1+1 < p+1 & p < 28+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 23;
      hence thesis by Ttool23a;
    end;
    suppose 23 <= p <= 23+1 or 24 <= p <= 24+1 or 25 <= p <= 25+1 or 
      26 <= p <= 26+1 or 27 <= p <= 27+1;
      then p = 23 by XPRIMES0:24,25,26,27,28,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 841 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
  proof
    assume p*p <= k < 841;
    then p*p < 29*29 by XXREAL_0:2;
    hence thesis by Ttool29a,NAT_4:1;
  end;

theorem Ttool31a:
  p < 31 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
  proof
    assume p < 31;
    then 1+1 < p+1 & p < 30+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 29;
      hence thesis by Ttool29a;
    end;
    suppose 29 <= p <= 29+1;
      hence thesis by XPRIMES0:30,NAT_1:9;
    end;
  end;

theorem
  p*p <= k < 961 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
  proof
    assume p*p <= k < 961;
    then p*p < 31*31 by XXREAL_0:2;
    hence thesis by Ttool31a,NAT_4:1;
  end;

theorem Ttool37a:
  p < 37 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
  proof
    assume p < 37;
    then 1+1 < p+1 & p < 36+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 31;
      hence thesis by Ttool31a;
    end;
    suppose 31 <= p <= 31+1 or 32 <= p <= 32+1 or 33 <= p <= 33+1 or 
      34 <= p <= 34+1 or 35 <= p <= 35+1;
      then p = 31 by XPRIMES0:32,33,34,35,36,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 1369 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
  proof
    assume p*p <= k < 1369;
    then p*p < 37*37 by XXREAL_0:2;
    hence thesis by Ttool37a,NAT_4:1;
  end;

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;

theorem
  p*p <= k < 1681 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*p <= k < 1681;
    then p*p < 41*41 by XXREAL_0:2;
    hence thesis by Ttool41a,NAT_4:1;
  end;

theorem Ttool43a:
  p < 43 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
  proof
    assume p < 43;
    then 1+1 < p+1 & p < 42+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 41;
      hence thesis by Ttool41a;
    end;
    suppose 41 <= p <= 41+1;
      then p = 41 by XPRIMES0:42,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 1849 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
  proof
    assume p*p <= k < 1849;
    then p*p < 43*43 by XXREAL_0:2;
    hence thesis by Ttool43a,NAT_4:1;
  end;

theorem Ttool47a:
  p < 47 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
  proof
    assume p < 47;
    then 1+1 < p+1 & p < 46+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 43;
      hence thesis by Ttool43a;
    end;
    suppose 43 <= p <= 43+1 or 44 <= p <= 44+1 or 45 <= p <= 45+1;
      then p = 43 by XPRIMES0:44,45,46,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 2209 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
  proof
    assume p*p <= k < 2209;
    then p*p < 47*47 by XXREAL_0:2;
    hence thesis by Ttool47a,NAT_4:1;
  end;

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;

theorem
  p*p <= k < 2809 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*p <= k < 2809;
    then p*p < 53*53 by XXREAL_0:2;
    hence thesis by Ttool53a,NAT_4:1;
  end;

theorem Ttool59a:
  p < 59 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
  proof
    assume p < 59;
    then 1+1 < p+1 & p < 58+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 53;
      hence thesis by Ttool53a;
    end;
    suppose 53 <= p <= 53+1 or 54 <= p <= 54+1 or 55 <= p <= 55+1 or 
      56 <= p <= 56+1 or 57 <= p <= 57+1;
      then p = 53 by XPRIMES0:54,55,56,57,58,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 3481 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
  proof
    assume p*p <= k < 3481;
    then p*p < 59*59 by XXREAL_0:2;
    hence thesis by Ttool59a,NAT_4:1;
  end;

theorem Ttool61a:
  p < 61 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
  proof
    assume p < 61;
    then 1+1 < p+1 & p < 60+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 59;
      hence thesis by Ttool59a;
    end;
    suppose 59 <= p <= 59+1;
      then p = 59 by XPRIMES0:60,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 3721 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
  proof
    assume p*p <= k < 3721;
    then p*p < 61*61 by XXREAL_0:2;
    hence thesis by Ttool61a,NAT_4:1;
  end;

theorem Ttool67a:
  p < 67 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
  proof
    assume p < 67;
    then 1+1 < p+1 & p < 66+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 61;
      hence thesis by Ttool61a;
    end;
    suppose 61 <= p <= 61+1 or 62 <= p <= 62+1 or 63 <= p <= 63+1 or 
      64 <= p <= 64+1 or 65 <= p <= 65+1;
      then p = 61 by XPRIMES0:62,63,64,65,66,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 4489 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
  proof
    assume p*p <= k < 4489;
    then p*p < 67*67 by XXREAL_0:2;
    hence thesis by Ttool67a,NAT_4:1;
  end;

theorem Ttool71a:
  p < 71 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
  proof
    assume p < 71;
    then 1+1 < p+1 & p < 70+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 67;
      hence thesis by Ttool67a;
    end;
    suppose 67 <= p <= 67+1 or 68 <= p <= 68+1 or 69 <= p <= 69+1;
      then p = 67 by XPRIMES0:68,69,70,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 5041 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
  proof
    assume p*p <= k < 5041;
    then p*p < 71*71 by XXREAL_0:2;
    hence thesis by Ttool71a,NAT_4:1;
  end;

theorem Ttool73a:
  p < 73 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
  proof
    assume p < 73;
    then 1+1 < p+1 & p < 72+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 71;
      hence thesis by Ttool71a;
    end;
    suppose 71 <= p <= 71+1;
      then p = 71 by XPRIMES0:72,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 5329 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
  proof
    assume p*p <= k < 5329;
    then p*p < 73*73 by XXREAL_0:2;
    hence thesis by Ttool73a,NAT_4:1;
  end;

theorem Ttool79a:
  p < 79 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
  proof
    assume p < 79;
    then 1+1 < p+1 & p < 78+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 73;
      hence thesis by Ttool73a;
    end;
    suppose 73 <= p <= 73+1 or 74 <= p <= 74+1 or 75 <= p <= 75+1 or 
      76 <= p <= 76+1 or 77 <= p <= 77+1;
      then p = 73 by XPRIMES0:74,75,76,77,78,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 6241 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
  proof
    assume p*p <= k < 6241;
    then p*p < 79*79 by XXREAL_0:2;
    hence thesis by Ttool79a,NAT_4:1;
  end;

theorem Ttool83a:
  p < 83 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
  proof
    assume p < 83;
    then 1+1 < p+1 & p < 82+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 79;
      hence thesis by Ttool79a;
    end;
    suppose 79 <= p <= 79+1 or 80 <= p <= 80+1 or 81 <= p <= 81+1;
      then p = 79 by XPRIMES0:80,81,82,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 6889 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
  proof
    assume p*p <= k < 6889;
    then p*p < 83*83 by XXREAL_0:2;
    hence thesis by Ttool83a,NAT_4:1;
  end;

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;

theorem
  p*p <= k < 7921 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*p <= k < 7921;
    then p*p < 89*89 by XXREAL_0:2;
    hence thesis by Ttool89a,NAT_4:1;
  end;

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;

theorem
  p*p <= k < 9409 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*p <= k < 9409;
    then p*p < 97*97 by XXREAL_0:2;
    hence thesis by Ttool97a,NAT_4:1;
  end;

theorem Ttool101a:
  p < 101 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
  proof
    assume p < 101;
    then 1+1 < p+1 & p < 100+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 97;
      hence thesis by Ttool97a;
    end;
    suppose 97 <= p <= 97+1 or 98 <= p <= 98+1 or 99 <= p <= 99+1;
      then p = 97 by XPRIMES0:100,98,99,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 10201 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
  proof
    assume p*p <= k < 10201;
    then p*p < 101*101 by XXREAL_0:2;
    hence thesis by Ttool101a,NAT_4:1;
  end;

theorem Ttool103a:
  p < 103 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
  proof
    assume p < 103;
    then 1+1 < p+1 & p < 102+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 101;
      hence thesis by Ttool101a;
    end;
    suppose 101 <= p <= 101+1;
      then p = 101 by XPRIMES0:102,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 10609 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
  proof
    assume p*p <= k < 10609;
    then p*p < 103*103 by XXREAL_0:2;
    hence thesis by Ttool103a,NAT_4:1;
  end;

theorem Ttool107a:
  p < 107 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
  proof
    assume p < 107;
    then 1+1 < p+1 & p < 106+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 103;
      hence thesis by Ttool103a;
    end;
    suppose 103 <= p <= 103+1 or 104 <= p <= 104+1 or 105 <= p <= 105+1;
      then p = 103 by XPRIMES0:104,105,106,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 11449 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
  proof
    assume p*p <= k < 11449;
    then p*p < 107*107 by XXREAL_0:2;
    hence thesis by Ttool107a,NAT_4:1;
  end;

theorem Ttool109a:
  p < 109 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
  proof
    assume p < 109;
    then 1+1 < p+1 & p < 108+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 107;
      hence thesis by Ttool107a;
    end;
    suppose 107 <= p <= 107+1;
      then p = 107 by XPRIMES0:108,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 11881 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
  proof
    assume p*p <= k < 11881;
    then p*p < 109*109 by XXREAL_0:2;
    hence thesis by Ttool109a,NAT_4:1;
  end;

theorem Ttool113a:
  p < 113 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
  proof
    assume p < 113;
    then 1+1 < p+1 & p < 112+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 109;
      hence thesis by Ttool109a;
    end;
    suppose 109 <= p <= 109+1 or 110 <= p <= 110+1 or 111 <= p <= 111+1;
      then p = 109 by XPRIMES0:110,111,112,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 12769 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
  proof
    assume p*p <= k < 12769;
    then p*p < 113*113 by XXREAL_0:2;
    hence thesis by Ttool113a,NAT_4:1;
  end;

theorem Ttool127a:
  p < 127 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
  proof
    assume p < 127;
    then 1+1 < p+1 & p < 126+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 113;
      hence thesis by Ttool113a;
    end;
    suppose 113 <= p & p <= 120;
      then 113 <= p <= 113+1 or 114 <= p <= 114+1 or 115 <= p <= 115+1 or 
        116 <= p <= 116+1 or 117 <= p <= 117+1 or 118 <= p <= 118+1 or 
        119 <= p <= 119+1;
      then p = 113 by XPRIMES0:114,115,116,117,118,119,120,NAT_1:9;
      hence thesis;
    end;
    suppose 120 <= p <= 120+1 or 121 <= p <= 121+1 or 122 <= p <= 122+1 or
      123 <= p <= 123+1 or 124 <= p <= 124+1 or 125 <= p <= 125+1;
      then contradiction by XPRIMES0:120,121,122,123,124,125,126,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 16129 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
  proof
    assume p*p <= k < 16129;
    then p*p < 127*127 by XXREAL_0:2;
    hence thesis by Ttool127a,NAT_4:1;
  end;

theorem Ttool131a:
  p < 131 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
  proof
    assume p < 131;
    then 1+1 < p+1 & p < 130+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 127;
      hence thesis by Ttool127a;
    end;
    suppose 127 <= p <= 127+1 or 128 <= p <= 128+1 or 129 <= p <= 129+1;
      then p = 127 by XPRIMES0:128,129,130,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 17161 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
  proof
    assume p*p <= k < 17161;
    then p*p < 131*131 by XXREAL_0:2;
    hence thesis by Ttool131a,NAT_4:1;
  end;

theorem Ttool137a:
  p < 137 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
  proof
    assume p < 137;
    then 1+1 < p+1 & p < 136+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 131;
      hence thesis by Ttool131a;
    end;
    suppose 131 <= p <= 131+1 or 132 <= p <= 132+1 or 133 <= p <= 133+1 or 
      134 <= p <= 134+1 or 135 <= p <= 135+1;
      then p = 131 by XPRIMES0:132,133,134,135,136,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 18769 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
  proof
    assume p*p <= k < 18769;
    then p*p < 137*137 by XXREAL_0:2;
    hence thesis by Ttool137a,NAT_4:1;
  end;

theorem Ttool139a:
  p < 139 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
  proof
    assume p < 139;
    then 1+1 < p+1 & p < 138+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 137;
      hence thesis by Ttool137a;
    end;
    suppose 137 <= p <= 137+1;
      then p = 137 by XPRIMES0:138,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 19321 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
  proof
    assume p*p <= k < 19321;
    then p*p < 139*139 by XXREAL_0:2;
    hence thesis by Ttool139a,NAT_4:1;
  end;

theorem Ttool149a:
  p < 149 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
  proof
    assume p < 149;
    then 1+1 < p+1 & p < 148+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 139;
      hence thesis by Ttool139a;
    end;
    suppose 139 <= p <= 139+1 or 140 <= p <= 140+1 or 141 <= p <= 141+1 or 
      142 <= p <= 142+1 or 143 <= p <= 143+1 or 144 <= p <= 144+1 or 
      145 <= p <= 145+1 or 146 <= p <= 146+1 or 147 <= p <= 147+1;
      then p = 139 by XPRIMES0:140,141,142,143,144,145,146,147,148,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 22201 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
  proof
    assume p*p <= k < 22201;
    then p*p < 149*149 by XXREAL_0:2;
    hence thesis by Ttool149a,NAT_4:1;
  end;

theorem Ttool151a:
  p < 151 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
  proof
    assume p < 151;
    then 1+1 < p+1 & p < 150+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 149;
      hence thesis by Ttool149a;
    end;
    suppose 149 <= p <= 149+1;
      then p = 149 by XPRIMES0:150,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 22801 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
  proof
    assume p*p <= k < 22801;
    then p*p < 151*151 by XXREAL_0:2;
    hence thesis by Ttool151a,NAT_4:1;
  end;

theorem Ttool157a:
  p < 157 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
  proof
    assume p < 157;
    then 1+1 < p+1 & p < 156+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 151;
      hence thesis by Ttool151a;
    end;
    suppose 151 <= p <= 151+1 or 152 <= p <= 152+1 or 153 <= p <= 153+1 or 
      154 <= p <= 154+1 or 155 <= p <= 155+1;
      then p = 151 by XPRIMES0:152,153,154,155,156,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 24649 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
  proof
    assume p*p <= k < 24649;
    then p*p < 157*157 by XXREAL_0:2;
    hence thesis by Ttool157a,NAT_4:1;
  end;

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;

theorem
  p*p <= k < 26569 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*p <= k < 26569;
    then p*p < 163*163 by XXREAL_0:2;
    hence thesis by Ttool163a,NAT_4:1;
  end;

theorem Ttool167a:
  p < 167 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 or p = 163
  proof
    assume p < 167;
    then 1+1 < p+1 & p < 166+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 163;
      hence thesis by Ttool163a;
    end;
    suppose 163 <= p <= 163+1 or 164 <= p <= 164+1 or 165 <= p <= 165+1;
      then p = 163 by XPRIMES0:164,165,166,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 27889 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 or p = 163
  proof
    assume p*p <= k < 27889;
    then p*p < 167*167 by XXREAL_0:2;
    hence thesis by Ttool167a,NAT_4:1;
  end;

theorem Ttool173a:
  p < 173 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 or p = 163 or p = 167
  proof
    assume p < 173;
    then 1+1 < p+1 & p < 172+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 167;
      hence thesis by Ttool167a;
    end;
    suppose 167 <= p <= 167+1 or 168 <= p <= 168+1 or 169 <= p <= 169+1 or 
      170 <= p <= 170+1 or 171 <= p <= 171+1;
      then p = 167 by XPRIMES0:168,169,170,171,172,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 29929 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 or p = 163 or p = 167
  proof
    assume p*p <= k < 29929;
    then p*p < 173*173 by XXREAL_0:2;
    hence thesis by Ttool173a,NAT_4:1;
  end;

theorem Ttool179a:
  p < 179 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 or p = 163 or p = 167 or p = 173
  proof
    assume p < 179;
    then 1+1 < p+1 & p < 178+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 173;
      hence thesis by Ttool173a;
    end;
    suppose 173 <= p <= 173+1 or 174 <= p <= 174+1 or 175 <= p <= 175+1 or 
      176 <= p <= 176+1 or 177 <= p <= 177+1;
      then p = 173 by XPRIMES0:174,175,176,177,178,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 32041 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 or p = 163 or p = 167 or p = 173
  proof
    assume p*p <= k < 32041;
    then p*p < 179*179 by XXREAL_0:2;
    hence thesis by Ttool179a,NAT_4:1;
  end;

theorem Ttool181a:
  p < 181 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 or p = 163 or p = 167 or p = 173 or p = 179
  proof
    assume p < 181;
    then 1+1 < p+1 & p < 180+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 179;
      hence thesis by Ttool179a;
    end;
    suppose 179 <= p <= 179+1;
      then p = 179 by XPRIMES0:180,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 32761 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 or p = 163 or p = 167 or p = 173 or p = 179
  proof
    assume p*p <= k < 32761;
    then p*p < 181*181 by XXREAL_0:2;
    hence thesis by Ttool181a,NAT_4:1;
  end;

theorem Ttool191a:
  p < 191 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181
  proof
    assume p < 191;
    then 1+1 < p+1 & p < 190+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 181;
      hence thesis by Ttool181a;
    end;
    suppose 181 <= p <= 181+1 or 182 <= p <= 182+1 or 183 <= p <= 183+1 or 
      184 <= p <= 184+1 or 185 <= p <= 185+1 or 186 <= p <= 186+1 or 
      187 <= p <= 187+1 or 188 <= p <= 188+1 or 189 <= p <= 189+1;
      then p = 181 by XPRIMES0:182,183,184,185,186,187,188,189,190,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 36481 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181
  proof
    assume p*p <= k < 36481;
    then p*p < 191*191 by XXREAL_0:2;
    hence thesis by Ttool191a,NAT_4:1;
  end;

theorem Ttool193a:
  p < 193 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191
  proof
    assume p < 193;
    then 1+1 < p+1 & p < 192+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 191;
      hence thesis by Ttool191a;
    end;
    suppose 191 <= p <= 191+1;
      then p = 191 by XPRIMES0:192,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 37249 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191
  proof
    assume p*p <= k < 37249;
    then p*p < 193*193 by XXREAL_0:2;
    hence thesis by Ttool193a,NAT_4:1;
  end;

theorem Ttool197a:
  p < 197 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193
  proof
    assume p < 197;
    then 1+1 < p+1 & p < 196+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 193;
      hence thesis by Ttool193a;
    end;
    suppose 193 <= p <= 193+1 or 194 <= p <= 194+1 or 195 <= p <= 195+1;
      then p = 193 by XPRIMES0:194,195,196,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 38809 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193
  proof
    assume p*p <= k < 38809;
    then p*p < 197*197 by XXREAL_0:2;
    hence thesis by Ttool197a,NAT_4:1;
  end;

theorem Ttool199a:
  p < 199 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197
  proof
    assume p < 199;
    then 1+1 < p+1 & p < 198+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 197;
      hence thesis by Ttool197a;
    end;
    suppose 197 <= p <= 197+1;
      then p = 197 by XPRIMES0:198,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 39601 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197
  proof
    assume p*p <= k < 39601;
    then p*p < 199*199 by XXREAL_0:2;
    hence thesis by Ttool199a,NAT_4:1;
  end;

theorem Ttool211a:
  p < 211 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199
  proof
    assume p < 211;
    then 1+1 < p+1 & p < 210+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 199;
      hence thesis by Ttool199a;
    end;
    suppose 199 <= p <= 199+1 or 200 <= p <= 200+1 or 201 <= p <= 201+1 or 
      202 <= p <= 202+1 or 203 <= p <= 203+1 or 204 <= p <= 204+1 or 
      205 <= p <= 205+1 or 206 <= p <= 206+1 or 207 <= p <= 207+1 or 
      208 <= p <= 208+1 or 209 <= p <= 209+1;
      then p = 199 by XPRIMES0:200,201,202,203,204,205,206,207,208,209,210,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 44521 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199
  proof
    assume p*p <= k < 44521;
    then p*p < 211*211 by XXREAL_0:2;
    hence thesis by Ttool211a,NAT_4:1;
  end;

theorem Ttool223a:
  p < 223 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211
  proof
    assume p < 223;
    then 1+1 < p+1 & p < 222+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 211;
      hence thesis by Ttool211a;
    end;
    suppose 211 <= p <= 211+1 or 212 <= p <= 212+1 or 213 <= p <= 213+1 or 
      214 <= p <= 214+1 or 215 <= p <= 215+1 or 216 <= p <= 216+1 or 
      217 <= p <= 217+1 or 218 <= p <= 218+1 or 219 <= p <= 219+1 or 
      220 <= p <= 220+1 or 221 <= p <= 221+1;
      then p = 211 by XPRIMES0:212,213,214,215,216,217,218,219,220,221,222,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 49729 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211
  proof
    assume p*p <= k < 49729;
    then p*p < 223*223 by XXREAL_0:2;
    hence thesis by Ttool223a,NAT_4:1;
  end;

theorem Ttool227a:
  p < 227 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223
  proof
    assume p < 227;
    then 1+1 < p+1 & p < 226+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 223;
      hence thesis by Ttool223a;
    end;
    suppose 223 <= p <= 223+1 or 224 <= p <= 224+1 or 225 <= p <= 225+1;
      then p = 223 by XPRIMES0:224,225,226,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 51529 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223
  proof
    assume p*p <= k < 51529;
    then p*p < 227*227 by XXREAL_0:2;
    hence thesis by Ttool227a,NAT_4:1;
  end;

theorem Ttool229a:
  p < 229 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227
  proof
    assume p < 229;
    then 1+1 < p+1 & p < 228+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 227;
      hence thesis by Ttool227a;
    end;
    suppose 227 <= p <= 227+1;
      then p = 227 by XPRIMES0:228,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 52441 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227
  proof
    assume p*p <= k < 52441;
    then p*p < 229*229 by XXREAL_0:2;
    hence thesis by Ttool229a,NAT_4:1;
  end;

theorem Ttool233a:
  p < 233 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229
  proof
    assume p < 233;
    then 1+1 < p+1 & p < 232+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 229;
      hence thesis by Ttool229a;
    end;
    suppose 229 <= p <= 229+1 or 230 <= p <= 230+1 or 231 <= p <= 231+1;
      then p = 229 by XPRIMES0:230,231,232,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 54289 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229
  proof
    assume p*p <= k < 54289;
    then p*p < 233*233 by XXREAL_0:2;
    hence thesis by Ttool233a,NAT_4:1;
  end;

theorem Ttool239a:
  p < 239 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233
  proof
    assume p < 239;
    then 1+1 < p+1 & p < 238+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 233;
      hence thesis by Ttool233a;
    end;
    suppose 233 <= p <= 233+1 or 234 <= p <= 234+1 or 235 <= p <= 235+1 or 
      236 <= p <= 236+1 or 237 <= p <= 237+1;
      then p = 233 by XPRIMES0:234,235,236,237,238,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 57121 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233
  proof
    assume p*p <= k < 57121;
    then p*p < 239*239 by XXREAL_0:2;
    hence thesis by Ttool239a,NAT_4:1;
  end;

theorem Ttool241a:
  p < 241 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239
  proof
    assume p < 241;
    then 1+1 < p+1 & p < 240+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 239;
      hence thesis by Ttool239a;
    end;
    suppose 239 <= p <= 239+1;
      then p = 239 by XPRIMES0:240,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 58081 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239
  proof
    assume p*p <= k < 58081;
    then p*p < 241*241 by XXREAL_0:2;
    hence thesis by Ttool241a,NAT_4:1;
  end;

theorem Ttool251a:
  p < 251 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241
  proof
    assume p < 251;
    then 1+1 < p+1 & p < 250+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 241;
      hence thesis by Ttool241a;
    end;
    suppose 241 <= p <= 241+1 or 242 <= p <= 242+1 or 243 <= p <= 243+1 or 
      244 <= p <= 244+1 or 245 <= p <= 245+1 or 246 <= p <= 246+1 or 
      247 <= p <= 247+1 or 248 <= p <= 248+1 or 249 <= p <= 249+1;
      then p = 241 by XPRIMES0:242,243,244,245,246,247,248,249,250,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 63001 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241
  proof
    assume p*p <= k < 63001;
    then p*p < 251*251 by XXREAL_0:2;
    hence thesis by Ttool251a,NAT_4:1;
  end;

theorem Ttool257a:
  p < 257 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251
  proof
    assume p < 257;
    then 1+1 < p+1 & p < 256+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 251;
      hence thesis by Ttool251a;
    end;
    suppose 251 <= p <= 251+1 or 252 <= p <= 252+1 or 253 <= p <= 253+1 or 
      254 <= p <= 254+1 or 255 <= p <= 255+1;
      then p = 251 by XPRIMES0:252,253,254,255,256,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 66049 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251
  proof
    assume p*p <= k < 66049;
    then p*p < 257*257 by XXREAL_0:2;
    hence thesis by Ttool257a,NAT_4:1;
  end;

theorem Ttool263a:
  p < 263 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257
  proof
    assume p < 263;
    then 1+1 < p+1 & p < 262+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 257;
      hence thesis by Ttool257a;
    end;
    suppose 257 <= p <= 257+1 or 258 <= p <= 258+1 or 259 <= p <= 259+1 or 
      260 <= p <= 260+1 or 261 <= p <= 261+1;
      then p = 257 by XPRIMES0:258,259,260,261,262,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 69169 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257
  proof
    assume p*p <= k < 69169;
    then p*p < 263*263 by XXREAL_0:2;
    hence thesis by Ttool263a,NAT_4:1;
  end;

theorem Ttool269a:
  p < 269 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263
  proof
    assume p < 269;
    then 1+1 < p+1 & p < 268+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 263;
      hence thesis by Ttool263a;
    end;
    suppose 263 <= p <= 263+1 or 264 <= p <= 264+1 or 265 <= p <= 265+1 or 
      266 <= p <= 266+1 or 267 <= p <= 267+1;
      then p = 263 by XPRIMES0:264,265,266,267,268,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 72361 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263
  proof
    assume p*p <= k < 72361;
    then p*p < 269*269 by XXREAL_0:2;
    hence thesis by Ttool269a,NAT_4:1;
  end;

theorem Ttool271a:
  p < 271 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269
  proof
    assume p < 271;
    then 1+1 < p+1 & p < 270+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 269;
      hence thesis by Ttool269a;
    end;
    suppose 269 <= p <= 269+1;
      then p = 269 by XPRIMES0:270,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 73441 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269
  proof
    assume p*p <= k < 73441;
    then p*p < 271*271 by XXREAL_0:2;
    hence thesis by Ttool271a,NAT_4:1;
  end;

theorem Ttool277a:
  p < 277 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271
  proof
    assume p < 277;
    then 1+1 < p+1 & p < 276+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 271;
      hence thesis by Ttool271a;
    end;
    suppose 271 <= p <= 271+1 or 272 <= p <= 272+1 or 273 <= p <= 273+1 or 
      274 <= p <= 274+1 or 275 <= p <= 275+1;
      then p = 271 by XPRIMES0:272,273,274,275,276,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 76729 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271
  proof
    assume p*p <= k < 76729;
    then p*p < 277*277 by XXREAL_0:2;
    hence thesis by Ttool277a,NAT_4:1;
  end;

theorem Ttool281a:
  p < 281 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277
  proof
    assume p < 281;
    then 1+1 < p+1 & p < 280+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 277;
      hence thesis by Ttool277a;
    end;
    suppose 277 <= p <= 277+1 or 278 <= p <= 278+1 or 279 <= p <= 279+1;
      then p = 277 by XPRIMES0:278,279,280,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 78961 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277
  proof
    assume p*p <= k < 78961;
    then p*p < 281*281 by XXREAL_0:2;
    hence thesis by Ttool281a,NAT_4:1;
  end;

theorem Ttool283a:
  p < 283 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281
  proof
    assume p < 283;
    then 1+1 < p+1 & p < 282+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 281;
      hence thesis by Ttool281a;
    end;
    suppose 281 <= p <= 281+1;
      then p = 281 by XPRIMES0:282,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 80089 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281
  proof
    assume p*p <= k < 80089;
    then p*p < 283*283 by XXREAL_0:2;
    hence thesis by Ttool283a,NAT_4:1;
  end;

theorem Ttool293a:
  p < 293 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283
  proof
    assume p < 293;
    then 1+1 < p+1 & p < 292+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 283;
      hence thesis by Ttool283a;
    end;
    suppose 283 <= p <= 283+1 or 284 <= p <= 284+1 or 285 <= p <= 285+1 or 
      286 <= p <= 286+1 or 287 <= p <= 287+1 or 288 <= p <= 288+1 or 
      289 <= p <= 289+1 or 290 <= p <= 290+1 or 291 <= p <= 291+1;
      then p = 283 by XPRIMES0:284,285,286,287,288,289,290,291,292,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 85849 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283
  proof
    assume p*p <= k < 85849;
    then p*p < 293*293 by XXREAL_0:2;
    hence thesis by Ttool293a,NAT_4:1;
  end;

theorem Ttool307a:
  p < 307 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293
  proof
    assume p < 307;
    then 1+1 < p+1 & p < 306+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 293;
      hence thesis by Ttool293a;
    end;
    suppose 293 <= p <= 299+1;
      then 293 <= p <= 293+1 or 294 <= p <= 294+1 or 295 <= p <= 295+1 or 
      296 <= p <= 296+1 or 297 <= p <= 297+1 or 298 <= p <= 298+1 or
      299 <= p <= 299+1;
      then p = 293 by XPRIMES0:294,295,296,297,298,299,300,NAT_1:9;
      hence thesis;
    end;
    suppose 300 <= p <= 300+1 or 301 <= p <= 301+1 or 
      302 <= p <= 302+1 or 303 <= p <= 303+1 or 304 <= p <= 304+1 or 
      305 <= p <= 305+1;
      then contradiction by XPRIMES0:300,301,302,303,304,305,306,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 94249 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293
  proof
    assume p*p <= k < 94249;
    then p*p < 307*307 by XXREAL_0:2;
    hence thesis by Ttool307a,NAT_4:1;
  end;

theorem Ttool311a:
  p < 311 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307
  proof
    assume p < 311;
    then 1+1 < p+1 & p < 310+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 307;
      hence thesis by Ttool307a;
    end;
    suppose 307 <= p <= 307+1 or 308 <= p <= 308+1 or 309 <= p <= 309+1;
      then p = 307 by XPRIMES0:308,309,310,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 96721 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307
  proof
    assume p*p <= k < 96721;
    then p*p < 311*311 by XXREAL_0:2;
    hence thesis by Ttool311a,NAT_4:1;
  end;

theorem Ttool313a:
  p < 313 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311
  proof
    assume p < 313;
    then 1+1 < p+1 & p < 312+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 311;
      hence thesis by Ttool311a;
    end;
    suppose 311 <= p <= 311+1;
      then p = 311 by XPRIMES0:312,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 97969 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311
  proof
    assume p*p <= k < 97969;
    then p*p < 313*313 by XXREAL_0:2;
    hence thesis by Ttool313a,NAT_4:1;
  end;

theorem Ttool317a:
  p < 317 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313
  proof
    assume p < 317;
    then 1+1 < p+1 & p < 316+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 313;
      hence thesis by Ttool313a;
    end;
    suppose 313 <= p <= 313+1 or 314 <= p <= 314+1 or 315 <= p <= 315+1;
      then p = 313 by XPRIMES0:314,315,316,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 100489 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313
  proof
    assume p*p <= k < 100489;
    then p*p < 317*317 by XXREAL_0:2;
    hence thesis by Ttool317a,NAT_4:1;
  end;

theorem Ttool331a:
  p < 331 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317
  proof
    assume p < 331;
    then 1+1 < p+1 & p < 330+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 317;
      hence thesis by Ttool317a;
    end;
    suppose 317 <= p <= 323+1;
      then 317 <= p <= 317+1 or 318 <= p <= 318+1 or 319 <= p <= 319+1 or 
      320 <= p <= 320+1 or 321 <= p <= 321+1 or 322 <= p <= 322+1 or 
      323 <= p <= 323+1;
      then p = 317 by XPRIMES0:318,319,320,321,322,323,324,NAT_1:9;
      hence thesis;
    end;
    suppose 324 <= p <= 324+1 or 325 <= p <= 325+1 or 
      326 <= p <= 326+1 or 327 <= p <= 327+1 or 328 <= p <= 328+1 or 
      329 <= p <= 329+1;
      then contradiction by XPRIMES0:324,325,326,327,328,329,330,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 109561 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317
  proof
    assume p*p <= k < 109561;
    then p*p < 331*331 by XXREAL_0:2;
    hence thesis by Ttool331a,NAT_4:1;
  end;

theorem Ttool337a:
  p < 337 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331
  proof
    assume p < 337;
    then 1+1 < p+1 & p < 336+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 331;
      hence thesis by Ttool331a;
    end;
    suppose 331 <= p <= 331+1 or 332 <= p <= 332+1 or 333 <= p <= 333+1 or 
      334 <= p <= 334+1 or 335 <= p <= 335+1;
      then p = 331 by XPRIMES0:332,333,334,335,336,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 113569 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331
  proof
    assume p*p <= k < 113569;
    then p*p < 337*337 by XXREAL_0:2;
    hence thesis by Ttool337a,NAT_4:1;
  end;

theorem Ttool347a:
  p < 347 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337
  proof
    assume p < 347;
    then 1+1 < p+1 & p < 346+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 337;
      hence thesis by Ttool337a;
    end;
    suppose 337 <= p <= 337+1 or 338 <= p <= 338+1 or 339 <= p <= 339+1 or 
      340 <= p <= 340+1 or 341 <= p <= 341+1 or 342 <= p <= 342+1 or 
      343 <= p <= 343+1 or 344 <= p <= 344+1 or 345 <= p <= 345+1;
      then p = 337 by XPRIMES0:338,339,340,341,342,343,344,345,346,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 120409 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337
  proof
    assume p*p <= k < 120409;
    then p*p < 347*347 by XXREAL_0:2;
    hence thesis by Ttool347a,NAT_4:1;
  end;

theorem Ttool349a:
  p < 349 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347
  proof
    assume p < 349;
    then 1+1 < p+1 & p < 348+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 347;
      hence thesis by Ttool347a;
    end;
    suppose 347 <= p <= 347+1;
      then p = 347 by XPRIMES0:348,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 121801 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347
  proof
    assume p*p <= k < 121801;
    then p*p < 349*349 by XXREAL_0:2;
    hence thesis by Ttool349a,NAT_4:1;
  end;

theorem Ttool353a:
  p < 353 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349
  proof
    assume p < 353;
    then 1+1 < p+1 & p < 352+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 349;
      hence thesis by Ttool349a;
    end;
    suppose 349 <= p <= 349+1 or 350 <= p <= 350+1 or 351 <= p <= 351+1;
      then p = 349 by XPRIMES0:350,351,352,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 124609 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349
  proof
    assume p*p <= k < 124609;
    then p*p < 353*353 by XXREAL_0:2;
    hence thesis by Ttool353a,NAT_4:1;
  end;

theorem Ttool359a:
  p < 359 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353
  proof
    assume p < 359;
    then 1+1 < p+1 & p < 358+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 353;
      hence thesis by Ttool353a;
    end;
    suppose 353 <= p <= 353+1 or 354 <= p <= 354+1 or 355 <= p <= 355+1 or 
      356 <= p <= 356+1 or 357 <= p <= 357+1;
      then p = 353 by XPRIMES0:354,355,356,357,358,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 128881 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353
  proof
    assume p*p <= k < 128881;
    then p*p < 359*359 by XXREAL_0:2;
    hence thesis by Ttool359a,NAT_4:1;
  end;

theorem Ttool367a:
  p < 367 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359
  proof
    assume p < 367;
    then 1+1 < p+1 & p < 366+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 359;
      hence thesis by Ttool359a;
    end;
    suppose 359 <= p <= 359+1 or 360 <= p <= 360+1 or 361 <= p <= 361+1 or 
      362 <= p <= 362+1 or 363 <= p <= 363+1 or 364 <= p <= 364+1 or 
      365 <= p <= 365+1;
      then p = 359 by XPRIMES0:360,361,362,363,364,365,366,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 134689 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359
  proof
    assume p*p <= k < 134689;
    then p*p < 367*367 by XXREAL_0:2;
    hence thesis by Ttool367a,NAT_4:1;
  end;

theorem Ttool373a:
  p < 373 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367
  proof
    assume p < 373;
    then 1+1 < p+1 & p < 372+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 367;
      hence thesis by Ttool367a;
    end;
    suppose 367 <= p <= 367+1 or 368 <= p <= 368+1 or 369 <= p <= 369+1 or 
      370 <= p <= 370+1 or 371 <= p <= 371+1;
      then p = 367 by XPRIMES0:368,369,370,371,372,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 139129 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367
  proof
    assume p*p <= k < 139129;
    then p*p < 373*373 by XXREAL_0:2;
    hence thesis by Ttool373a,NAT_4:1;
  end;

theorem Ttool379a:
  p < 379 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373
  proof
    assume p < 379;
    then 1+1 < p+1 & p < 378+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 373;
      hence thesis by Ttool373a;
    end;
    suppose 373 <= p <= 373+1 or 374 <= p <= 374+1 or 375 <= p <= 375+1 or 
      376 <= p <= 376+1 or 377 <= p <= 377+1;
      then p = 373 by XPRIMES0:374,375,376,377,378,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 143641 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373
  proof
    assume p*p <= k < 143641;
    then p*p < 379*379 by XXREAL_0:2;
    hence thesis by Ttool379a,NAT_4:1;
  end;

theorem Ttool383a:
  p < 383 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379
  proof
    assume p < 383;
    then 1+1 < p+1 & p < 382+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 379;
      hence thesis by Ttool379a;
    end;
    suppose 379 <= p <= 379+1 or 380 <= p <= 380+1 or 381 <= p <= 381+1;
      then p = 379 by XPRIMES0:380,381,382,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 146689 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379
  proof
    assume p*p <= k < 146689;
    then p*p < 383*383 by XXREAL_0:2;
    hence thesis by Ttool383a,NAT_4:1;
  end;

theorem Ttool389a:
  p < 389 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383
  proof
    assume p < 389;
    then 1+1 < p+1 & p < 388+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 383;
      hence thesis by Ttool383a;
    end;
    suppose 383 <= p <= 383+1 or 384 <= p <= 384+1 or 385 <= p <= 385+1 or 
      386 <= p <= 386+1 or 387 <= p <= 387+1;
      then p = 383 by XPRIMES0:384,385,386,387,388,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 151321 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383
  proof
    assume p*p <= k < 151321;
    then p*p < 389*389 by XXREAL_0:2;
    hence thesis by Ttool389a,NAT_4:1;
  end;

theorem Ttool397a:
  p < 397 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389
  proof
    assume p < 397;
    then 1+1 < p+1 & p < 396+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 389;
      hence thesis by Ttool389a;
    end;
    suppose 389 <= p <= 389+1 or 390 <= p <= 390+1 or 391 <= p <= 391+1 or 
      392 <= p <= 392+1 or 393 <= p <= 393+1 or 394 <= p <= 394+1 or 
      395 <= p <= 395+1;
      then p = 389 by XPRIMES0:390,391,392,393,394,395,396,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 157609 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389
  proof
    assume p*p <= k < 157609;
    then p*p < 397*397 by XXREAL_0:2;
    hence thesis by Ttool397a,NAT_4:1;
  end;

theorem Ttool401a:
  p < 401 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397
  proof
    assume p < 401;
    then 1+1 < p+1 & p < 400+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 397;
      hence thesis by Ttool397a;
    end;
    suppose 397 <= p <= 397+1 or 398 <= p <= 398+1 or 399 <= p <= 399+1;
      then p = 397 by XPRIMES0:398,399,400,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 160801 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397
  proof
    assume p*p <= k < 160801;
    then p*p < 401*401 by XXREAL_0:2;
    hence thesis by Ttool401a,NAT_4:1;
  end;

theorem Ttool409a:
  p < 409 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401
  proof
    assume p < 409;
    then 1+1 < p+1 & p < 408+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 401;
      hence thesis by Ttool401a;
    end;
    suppose 401 <= p <= 401+1 or 402 <= p <= 402+1 or 403 <= p <= 403+1 or 
      404 <= p <= 404+1 or 405 <= p <= 405+1 or 406 <= p <= 406+1 or 
      407 <= p <= 407+1;
      then p = 401 by XPRIMES0:402,403,404,405,406,407,408,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 167281 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401
  proof
    assume p*p <= k < 167281;
    then p*p < 409*409 by XXREAL_0:2;
    hence thesis by Ttool409a,NAT_4:1;
  end;

theorem Ttool419a:
  p < 419 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409
  proof
    assume p < 419;
    then 1+1 < p+1 & p < 418+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 409;
      hence thesis by Ttool409a;
    end;
    suppose 409 <= p <= 409+1 or 410 <= p <= 410+1 or 411 <= p <= 411+1 or 
      412 <= p <= 412+1 or 413 <= p <= 413+1 or 414 <= p <= 414+1 or 
      415 <= p <= 415+1 or 416 <= p <= 416+1 or 417 <= p <= 417+1;
      then p = 409 by XPRIMES0:410,411,412,413,414,415,416,417,418,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 175561 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409
  proof
    assume p*p <= k < 175561;
    then p*p < 419*419 by XXREAL_0:2;
    hence thesis by Ttool419a,NAT_4:1;
  end;

theorem Ttool421a:
  p < 421 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419
  proof
    assume p < 421;
    then 1+1 < p+1 & p < 420+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 419;
      hence thesis by Ttool419a;
    end;
    suppose 419 <= p <= 419+1;
      then p = 419 by XPRIMES0:420,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 177241 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419
  proof
    assume p*p <= k < 177241;
    then p*p < 421*421 by XXREAL_0:2;
    hence thesis by Ttool421a,NAT_4:1;
  end;

theorem Ttool431a:
  p < 431 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421
  proof
    assume p < 431;
    then 1+1 < p+1 & p < 430+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 421;
      hence thesis by Ttool421a;
    end;
    suppose 421 <= p <= 421+1 or 422 <= p <= 422+1 or 423 <= p <= 423+1 or 
      424 <= p <= 424+1 or 425 <= p <= 425+1 or 426 <= p <= 426+1 or 
      427 <= p <= 427+1 or 428 <= p <= 428+1 or 429 <= p <= 429+1;
      then p = 421 by XPRIMES0:422,423,424,425,426,427,428,429,430,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 185761 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421
  proof
    assume p*p <= k < 185761;
    then p*p < 431*431 by XXREAL_0:2;
    hence thesis by Ttool431a,NAT_4:1;
  end;

theorem Ttool433a:
  p < 433 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431
  proof
    assume p < 433;
    then 1+1 < p+1 & p < 432+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 431;
      hence thesis by Ttool431a;
    end;
    suppose 431 <= p <= 431+1;
      then p = 431 by XPRIMES0:432,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 187489 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431
  proof
    assume p*p <= k < 187489;
    then p*p < 433*433 by XXREAL_0:2;
    hence thesis by Ttool433a,NAT_4:1;
  end;

theorem Ttool439a:
  p < 439 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433
  proof
    assume p < 439;
    then 1+1 < p+1 & p < 438+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 433;
      hence thesis by Ttool433a;
    end;
    suppose 433 <= p <= 433+1 or 434 <= p <= 434+1 or 435 <= p <= 435+1 or 
      436 <= p <= 436+1 or 437 <= p <= 437+1;
      then p = 433 by XPRIMES0:434,435,436,437,438,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 192721 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433
  proof
    assume p*p <= k < 192721;
    then p*p < 439*439 by XXREAL_0:2;
    hence thesis by Ttool439a,NAT_4:1;
  end;

theorem Ttool443a:
  p < 443 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439
  proof
    assume p < 443;
    then 1+1 < p+1 & p < 442+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 439;
      hence thesis by Ttool439a;
    end;
    suppose 439 <= p <= 439+1 or 440 <= p <= 440+1 or 441 <= p <= 441+1;
      then p = 439 by XPRIMES0:440,441,442,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 196249 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439
  proof
    assume p*p <= k < 196249;
    then p*p < 443*443 by XXREAL_0:2;
    hence thesis by Ttool443a,NAT_4:1;
  end;

theorem Ttool449a:
  p < 449 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443
  proof
    assume p < 449;
    then 1+1 < p+1 & p < 448+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 443;
      hence thesis by Ttool443a;
    end;
    suppose 443 <= p <= 443+1 or 444 <= p <= 444+1 or 445 <= p <= 445+1 or 
      446 <= p <= 446+1 or 447 <= p <= 447+1;
      then p = 443 by XPRIMES0:444,445,446,447,448,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 201601 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443
  proof
    assume p*p <= k < 201601;
    then p*p < 449*449 by XXREAL_0:2;
    hence thesis by Ttool449a,NAT_4:1;
  end;

theorem Ttool457a:
  p < 457 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449
  proof
    assume p < 457;
    then 1+1 < p+1 & p < 456+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 449;
      hence thesis by Ttool449a;
    end;
    suppose 449 <= p <= 449+1 or 450 <= p <= 450+1 or 451 <= p <= 451+1 or 
      452 <= p <= 452+1 or 453 <= p <= 453+1 or 454 <= p <= 454+1 or 
      455 <= p <= 455+1;
      then p = 449 by XPRIMES0:450,451,452,453,454,455,456,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 208849 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449
  proof
    assume p*p <= k < 208849;
    then p*p < 457*457 by XXREAL_0:2;
    hence thesis by Ttool457a,NAT_4:1;
  end;

theorem Ttool461a:
  p < 461 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457
  proof
    assume p < 461;
    then 1+1 < p+1 & p < 460+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 457;
      hence thesis by Ttool457a;
    end;
    suppose 457 <= p <= 457+1 or 458 <= p <= 458+1 or 459 <= p <= 459+1;
      then p = 457 by XPRIMES0:458,459,460,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 212521 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457
  proof
    assume p*p <= k < 212521;
    then p*p < 461*461 by XXREAL_0:2;
    hence thesis by Ttool461a,NAT_4:1;
  end;

theorem Ttool463a:
  p < 463 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461
  proof
    assume p < 463;
    then 1+1 < p+1 & p < 462+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 461;
      hence thesis by Ttool461a;
    end;
    suppose 461 <= p <= 461+1;
      then p = 461 by XPRIMES0:462,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 214369 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461
  proof
    assume p*p <= k < 214369;
    then p*p < 463*463 by XXREAL_0:2;
    hence thesis by Ttool463a,NAT_4:1;
  end;

theorem Ttool467a:
  p < 467 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463
  proof
    assume p < 467;
    then 1+1 < p+1 & p < 466+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 463;
      hence thesis by Ttool463a;
    end;
    suppose 463 <= p <= 463+1 or 464 <= p <= 464+1 or 465 <= p <= 465+1;
      then p = 463 by XPRIMES0:464,465,466,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 218089 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463
  proof
    assume p*p <= k < 218089;
    then p*p < 467*467 by XXREAL_0:2;
    hence thesis by Ttool467a,NAT_4:1;
  end;

theorem Ttool479a:
  p < 479 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467
  proof
    assume p < 479;
    then 1+1 < p+1 & p < 478+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 467;
      hence thesis by Ttool467a;
    end;
    suppose 467 <= p <= 467+1 or 468 <= p <= 468+1 or 469 <= p <= 469+1 or 
      470 <= p <= 470+1 or 471 <= p <= 471+1 or 472 <= p <= 472+1 or 
      473 <= p <= 473+1 or 474 <= p <= 474+1 or 475 <= p <= 475+1 or 
      476 <= p <= 476+1 or 477 <= p <= 477+1;
      then p = 467 by XPRIMES0:468,469,470,471,472,473,474,475,476,477,478,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 229441 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467
  proof
    assume p*p <= k < 229441;
    then p*p < 479*479 by XXREAL_0:2;
    hence thesis by Ttool479a,NAT_4:1;
  end;

theorem Ttool487a:
  p < 487 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479
  proof
    assume p < 487;
    then 1+1 < p+1 & p < 486+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 479;
      hence thesis by Ttool479a;
    end;
    suppose 479 <= p <= 479+1 or 480 <= p <= 480+1 or 481 <= p <= 481+1 or 
      482 <= p <= 482+1 or 483 <= p <= 483+1 or 484 <= p <= 484+1 or 
      485 <= p <= 485+1;
      then p = 479 by XPRIMES0:480,481,482,483,484,485,486,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 237169 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479
  proof
    assume p*p <= k < 237169;
    then p*p < 487*487 by XXREAL_0:2;
    hence thesis by Ttool487a,NAT_4:1;
  end;

theorem Ttool491a:
  p < 491 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487
  proof
    assume p < 491;
    then 1+1 < p+1 & p < 490+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 487;
      hence thesis by Ttool487a;
    end;
    suppose 487 <= p <= 487+1 or 488 <= p <= 488+1 or 489 <= p <= 489+1;
      then p = 487 by XPRIMES0:488,489,490,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 241081 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487
  proof
    assume p*p <= k < 241081;
    then p*p < 491*491 by XXREAL_0:2;
    hence thesis by Ttool491a,NAT_4:1;
  end;

theorem Ttool499a:
  p < 499 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491
  proof
    assume p < 499;
    then 1+1 < p+1 & p < 498+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 491;
      hence thesis by Ttool491a;
    end;
    suppose 491 <= p <= 491+1 or 492 <= p <= 492+1 or 493 <= p <= 493+1 or 
      494 <= p <= 494+1 or 495 <= p <= 495+1 or 496 <= p <= 496+1 or 
      497 <= p <= 497+1;
      then p = 491 by XPRIMES0:492,493,494,495,496,497,498,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 249001 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491
  proof
    assume p*p <= k < 249001;
    then p*p < 499*499 by XXREAL_0:2;
    hence thesis by Ttool499a,NAT_4:1;
  end;

theorem Ttool503a:
  p < 503 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499
  proof
    assume p < 503;
    then 1+1 < p+1 & p < 502+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 499;
      hence thesis by Ttool499a;
    end;
    suppose 499 <= p <= 499+1 or 500 <= p <= 500+1 or 501 <= p <= 501+1;
      then p = 499 by XPRIMES0:500,501,502,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 253009 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499
  proof
    assume p*p <= k < 253009;
    then p*p < 503*503 by XXREAL_0:2;
    hence thesis by Ttool503a,NAT_4:1;
  end;

theorem Ttool509a:
  p < 509 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503
  proof
    assume p < 509;
    then 1+1 < p+1 & p < 508+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 503;
      hence thesis by Ttool503a;
    end;
    suppose 503 <= p <= 503+1 or 504 <= p <= 504+1 or 505 <= p <= 505+1 or 
      506 <= p <= 506+1 or 507 <= p <= 507+1;
      then p = 503 by XPRIMES0:504,505,506,507,508,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 259081 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503
  proof
    assume p*p <= k < 259081;
    then p*p < 509*509 by XXREAL_0:2;
    hence thesis by Ttool509a,NAT_4:1;
  end;

theorem Ttool521a:
  p < 521 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509
  proof
    assume p < 521;
    then 1+1 < p+1 & p < 520+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 509;
      hence thesis by Ttool509a;
    end;
    suppose 509 <= p <= 509+1 or 510 <= p <= 510+1 or 511 <= p <= 511+1 or 
      512 <= p <= 512+1 or 513 <= p <= 513+1 or 514 <= p <= 514+1 or 
      515 <= p <= 515+1 or 516 <= p <= 516+1 or 517 <= p <= 517+1 or 
      518 <= p <= 518+1 or 519 <= p <= 519+1;
      then p = 509 by XPRIMES0:510,511,512,513,514,515,516,517,518,519,520,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 271441 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509
  proof
    assume p*p <= k < 271441;
    then p*p < 521*521 by XXREAL_0:2;
    hence thesis by Ttool521a,NAT_4:1;
  end;

theorem Ttool523a:
  p < 523 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521
  proof
    assume p < 523;
    then 1+1 < p+1 & p < 522+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 521;
      hence thesis by Ttool521a;
    end;
    suppose 521 <= p <= 521+1;
      then p = 521 by XPRIMES0:522,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 273529 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521
  proof
    assume p*p <= k < 273529;
    then p*p < 523*523 by XXREAL_0:2;
    hence thesis by Ttool523a,NAT_4:1;
  end;

theorem Ttool541a:
  p < 541 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523
  proof
    assume p < 541;
    then 1+1 < p+1 & p < 540+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 523;
      hence thesis by Ttool523a;
    end;
    suppose 523 <= p <= 531+1;
      then 523 <= p <= 523+1 or 524 <= p <= 524+1 or 525 <= p <= 525+1 or 
      526 <= p <= 526+1 or 527 <= p <= 527+1 or 528 <= p <= 528+1 or 
      529 <= p <= 529+1 or 530 <= p <= 530+1 or 531 <= p <= 531+1;
      then p = 523 by XPRIMES0:524,525,526,527,528,529,530,531,532,NAT_1:9;
      hence thesis;
    end;
    suppose 532 <= p <= 532+1 or 533 <= p <= 533+1 or 534 <= p <= 534+1 or 
      535 <= p <= 535+1 or 536 <= p <= 536+1 or 537 <= p <= 537+1 or 
      538 <= p <= 538+1 or 539 <= p <= 539+1;
      then contradiction by XPRIMES0:532,533,534,535,536,537,538,539,540,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 292681 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523
  proof
    assume p*p <= k < 292681;
    then p*p < 541*541 by XXREAL_0:2;
    hence thesis by Ttool541a,NAT_4:1;
  end;

theorem Ttool547a:
  p < 547 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541
  proof
    assume p < 547;
    then 1+1 < p+1 & p < 546+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 541;
      hence thesis by Ttool541a;
    end;
    suppose 541 <= p <= 541+1 or 542 <= p <= 542+1 or 543 <= p <= 543+1 or 
      544 <= p <= 544+1 or 545 <= p <= 545+1;
      then p = 541 by XPRIMES0:542,543,544,545,546,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 299209 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541
  proof
    assume p*p <= k < 299209;
    then p*p < 547*547 by XXREAL_0:2;
    hence thesis by Ttool547a,NAT_4:1;
  end;

theorem Ttool557a:
  p < 557 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547
  proof
    assume p < 557;
    then 1+1 < p+1 & p < 556+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 547;
      hence thesis by Ttool547a;
    end;
    suppose 547 <= p <= 547+1 or 548 <= p <= 548+1 or 549 <= p <= 549+1 or 
      550 <= p <= 550+1 or 551 <= p <= 551+1 or 552 <= p <= 552+1 or 
      553 <= p <= 553+1 or 554 <= p <= 554+1 or 555 <= p <= 555+1;
      then p = 547 by XPRIMES0:548,549,550,551,552,553,554,555,556,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 310249 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547
  proof
    assume p*p <= k < 310249;
    then p*p < 557*557 by XXREAL_0:2;
    hence thesis by Ttool557a,NAT_4:1;
  end;

theorem Ttool563a:
  p < 563 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557
  proof
    assume p < 563;
    then 1+1 < p+1 & p < 562+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 557;
      hence thesis by Ttool557a;
    end;
    suppose 557 <= p <= 557+1 or 558 <= p <= 558+1 or 559 <= p <= 559+1 or 
      560 <= p <= 560+1 or 561 <= p <= 561+1;
      then p = 557 by XPRIMES0:558,559,560,561,562,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 316969 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557
  proof
    assume p*p <= k < 316969;
    then p*p < 563*563 by XXREAL_0:2;
    hence thesis by Ttool563a,NAT_4:1;
  end;

theorem Ttool569a:
  p < 569 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563
  proof
    assume p < 569;
    then 1+1 < p+1 & p < 568+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 563;
      hence thesis by Ttool563a;
    end;
    suppose 563 <= p <= 563+1 or 564 <= p <= 564+1 or 565 <= p <= 565+1 or 
      566 <= p <= 566+1 or 567 <= p <= 567+1;
      then p = 563 by XPRIMES0:564,565,566,567,568,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 323761 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563
  proof
    assume p*p <= k < 323761;
    then p*p < 569*569 by XXREAL_0:2;
    hence thesis by Ttool569a,NAT_4:1;
  end;

theorem Ttool571a:
  p < 571 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569
  proof
    assume p < 571;
    then 1+1 < p+1 & p < 570+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 569;
      hence thesis by Ttool569a;
    end;
    suppose 569 <= p <= 569+1;
      then p = 569 by XPRIMES0:570,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 326041 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569
  proof
    assume p*p <= k < 326041;
    then p*p < 571*571 by XXREAL_0:2;
    hence thesis by Ttool571a,NAT_4:1;
  end;

theorem Ttool577a:
  p < 577 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571
  proof
    assume p < 577;
    then 1+1 < p+1 & p < 576+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 571;
      hence thesis by Ttool571a;
    end;
    suppose 571 <= p <= 571+1 or 572 <= p <= 572+1 or 573 <= p <= 573+1 or 
      574 <= p <= 574+1 or 575 <= p <= 575+1;
      then p = 571 by XPRIMES0:572,573,574,575,576,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 332929 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571
  proof
    assume p*p <= k < 332929;
    then p*p < 577*577 by XXREAL_0:2;
    hence thesis by Ttool577a,NAT_4:1;
  end;

theorem Ttool587a:
  p < 587 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577
  proof
    assume p < 587;
    then 1+1 < p+1 & p < 586+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 577;
      hence thesis by Ttool577a;
    end;
    suppose 577 <= p <= 577+1 or 578 <= p <= 578+1 or 579 <= p <= 579+1 or 
      580 <= p <= 580+1 or 581 <= p <= 581+1 or 582 <= p <= 582+1 or 
      583 <= p <= 583+1 or 584 <= p <= 584+1 or 585 <= p <= 585+1;
      then p = 577 by XPRIMES0:578,579,580,581,582,583,584,585,586,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 344569 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577
  proof
    assume p*p <= k < 344569;
    then p*p < 587*587 by XXREAL_0:2;
    hence thesis by Ttool587a,NAT_4:1;
  end;

theorem Ttool593a:
  p < 593 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587
  proof
    assume p < 593;
    then 1+1 < p+1 & p < 592+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 587;
      hence thesis by Ttool587a;
    end;
    suppose 587 <= p <= 587+1 or 588 <= p <= 588+1 or 589 <= p <= 589+1 or 
      590 <= p <= 590+1 or 591 <= p <= 591+1;
      then p = 587 by XPRIMES0:588,589,590,591,592,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 351649 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587
  proof
    assume p*p <= k < 351649;
    then p*p < 593*593 by XXREAL_0:2;
    hence thesis by Ttool593a,NAT_4:1;
  end;

theorem Ttool599a:
  p < 599 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593
  proof
    assume p < 599;
    then 1+1 < p+1 & p < 598+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 593;
      hence thesis by Ttool593a;
    end;
    suppose 593 <= p <= 593+1 or 594 <= p <= 594+1 or 595 <= p <= 595+1 or 
      596 <= p <= 596+1 or 597 <= p <= 597+1;
      then p = 593 by XPRIMES0:594,595,596,597,598,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 358801 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593
  proof
    assume p*p <= k < 358801;
    then p*p < 599*599 by XXREAL_0:2;
    hence thesis by Ttool599a,NAT_4:1;
  end;

theorem Ttool601a:
  p < 601 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599
  proof
    assume p < 601;
    then 1+1 < p+1 & p < 600+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 599;
      hence thesis by Ttool599a;
    end;
    suppose 599 <= p <= 599+1;
      then p = 599 by XPRIMES0:600,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 361201 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599
  proof
    assume p*p <= k < 361201;
    then p*p < 601*601 by XXREAL_0:2;
    hence thesis by Ttool601a,NAT_4:1;
  end;

theorem Ttool607a:
  p < 607 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601
  proof
    assume p < 607;
    then 1+1 < p+1 & p < 606+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 601;
      hence thesis by Ttool601a;
    end;
    suppose 601 <= p <= 601+1 or 602 <= p <= 602+1 or 603 <= p <= 603+1 or 
      604 <= p <= 604+1 or 605 <= p <= 605+1;
      then p = 601 by XPRIMES0:602,603,604,605,606,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 368449 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601
  proof
    assume p*p <= k < 368449;
    then p*p < 607*607 by XXREAL_0:2;
    hence thesis by Ttool607a,NAT_4:1;
  end;

theorem Ttool613a:
  p < 613 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607
  proof
    assume p < 613;
    then 1+1 < p+1 & p < 612+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 607;
      hence thesis by Ttool607a;
    end;
    suppose 607 <= p <= 607+1 or 608 <= p <= 608+1 or 609 <= p <= 609+1 or 
      610 <= p <= 610+1 or 611 <= p <= 611+1;
      then p = 607 by XPRIMES0:608,609,610,611,612,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 375769 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607
  proof
    assume p*p <= k < 375769;
    then p*p < 613*613 by XXREAL_0:2;
    hence thesis by Ttool613a,NAT_4:1;
  end;

theorem Ttool617a:
  p < 617 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613
  proof
    assume p < 617;
    then 1+1 < p+1 & p < 616+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 613;
      hence thesis by Ttool613a;
    end;
    suppose 613 <= p <= 613+1 or 614 <= p <= 614+1 or 615 <= p <= 615+1;
      then p = 613 by XPRIMES0:614,615,616,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 380689 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613
  proof
    assume p*p <= k < 380689;
    then p*p < 617*617 by XXREAL_0:2;
    hence thesis by Ttool617a,NAT_4:1;
  end;

theorem Ttool619a:
  p < 619 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617
  proof
    assume p < 619;
    then 1+1 < p+1 & p < 618+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 617;
      hence thesis by Ttool617a;
    end;
    suppose 617 <= p <= 617+1;
      then p = 617 by XPRIMES0:618,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 383161 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617
  proof
    assume p*p <= k < 383161;
    then p*p < 619*619 by XXREAL_0:2;
    hence thesis by Ttool619a,NAT_4:1;
  end;

theorem Ttool631a:
  p < 631 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619
  proof
    assume p < 631;
    then 1+1 < p+1 & p < 630+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 619;
      hence thesis by Ttool619a;
    end;
    suppose 619 <= p <= 619+1 or 620 <= p <= 620+1 or 621 <= p <= 621+1 or 
      622 <= p <= 622+1 or 623 <= p <= 623+1 or 624 <= p <= 624+1 or 
      625 <= p <= 625+1 or 626 <= p <= 626+1 or 627 <= p <= 627+1 or 
      628 <= p <= 628+1 or 629 <= p <= 629+1;
      then p = 619 by XPRIMES0:620,621,622,623,624,625,626,627,628,629,630,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 398161 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619
  proof
    assume p*p <= k < 398161;
    then p*p < 631*631 by XXREAL_0:2;
    hence thesis by Ttool631a,NAT_4:1;
  end;

theorem Ttool641a:
  p < 641 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631
  proof
    assume p < 641;
    then 1+1 < p+1 & p < 640+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 631;
      hence thesis by Ttool631a;
    end;
    suppose 631 <= p <= 631+1 or 632 <= p <= 632+1 or 633 <= p <= 633+1 or 
      634 <= p <= 634+1 or 635 <= p <= 635+1 or 636 <= p <= 636+1 or 
      637 <= p <= 637+1 or 638 <= p <= 638+1 or 639 <= p <= 639+1;
      then p = 631 by XPRIMES0:632,633,634,635,636,637,638,639,640,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 410881 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631
  proof
    assume p*p <= k < 410881;
    then p*p < 641*641 by XXREAL_0:2;
    hence thesis by Ttool641a,NAT_4:1;
  end;

theorem Ttool643a:
  p < 643 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641
  proof
    assume p < 643;
    then 1+1 < p+1 & p < 642+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 641;
      hence thesis by Ttool641a;
    end;
    suppose 641 <= p <= 641+1;
      then p = 641 by XPRIMES0:642,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 413449 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641
  proof
    assume p*p <= k < 413449;
    then p*p < 643*643 by XXREAL_0:2;
    hence thesis by Ttool643a,NAT_4:1;
  end;

theorem Ttool647a:
  p < 647 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643
  proof
    assume p < 647;
    then 1+1 < p+1 & p < 646+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 643;
      hence thesis by Ttool643a;
    end;
    suppose 643 <= p <= 643+1 or 644 <= p <= 644+1 or 645 <= p <= 645+1;
      then p = 643 by XPRIMES0:644,645,646,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 418609 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643
  proof
    assume p*p <= k < 418609;
    then p*p < 647*647 by XXREAL_0:2;
    hence thesis by Ttool647a,NAT_4:1;
  end;

theorem Ttool653a:
  p < 653 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647
  proof
    assume p < 653;
    then 1+1 < p+1 & p < 652+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 647;
      hence thesis by Ttool647a;
    end;
    suppose 647 <= p <= 647+1 or 648 <= p <= 648+1 or 649 <= p <= 649+1 or 
      650 <= p <= 650+1 or 651 <= p <= 651+1;
      then p = 647 by XPRIMES0:648,649,650,651,652,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 426409 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647
  proof
    assume p*p <= k < 426409;
    then p*p < 653*653 by XXREAL_0:2;
    hence thesis by Ttool653a,NAT_4:1;
  end;

theorem Ttool659a:
  p < 659 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653
  proof
    assume p < 659;
    then 1+1 < p+1 & p < 658+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 653;
      hence thesis by Ttool653a;
    end;
    suppose 653 <= p <= 653+1 or 654 <= p <= 654+1 or 655 <= p <= 655+1 or 
      656 <= p <= 656+1 or 657 <= p <= 657+1;
      then p = 653 by XPRIMES0:654,655,656,657,658,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 434281 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653
  proof
    assume p*p <= k < 434281;
    then p*p < 659*659 by XXREAL_0:2;
    hence thesis by Ttool659a,NAT_4:1;
  end;

theorem Ttool661a:
  p < 661 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659
  proof
    assume p < 661;
    then 1+1 < p+1 & p < 660+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 659;
      hence thesis by Ttool659a;
    end;
    suppose 659 <= p <= 659+1;
      then p = 659 by XPRIMES0:660,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 436921 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659
  proof
    assume p*p <= k < 436921;
    then p*p < 661*661 by XXREAL_0:2;
    hence thesis by Ttool661a,NAT_4:1;
  end;

theorem Ttool673a:
  p < 673 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661
  proof
    assume p < 673;
    then 1+1 < p+1 & p < 672+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 661;
      hence thesis by Ttool661a;
    end;
    suppose 661 <= p <= 661+1 or 662 <= p <= 662+1 or 663 <= p <= 663+1 or 
      664 <= p <= 664+1 or 665 <= p <= 665+1 or 666 <= p <= 666+1 or 
      667 <= p <= 667+1 or 668 <= p <= 668+1 or 669 <= p <= 669+1 or 
      670 <= p <= 670+1 or 671 <= p <= 671+1;
      then p = 661 by XPRIMES0:662,663,664,665,666,667,668,669,670,671,672,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 452929 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661
  proof
    assume p*p <= k < 452929;
    then p*p < 673*673 by XXREAL_0:2;
    hence thesis by Ttool673a,NAT_4:1;
  end;

theorem Ttool677a:
  p < 677 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673
  proof
    assume p < 677;
    then 1+1 < p+1 & p < 676+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 673;
      hence thesis by Ttool673a;
    end;
    suppose 673 <= p <= 673+1 or 674 <= p <= 674+1 or 675 <= p <= 675+1;
      then p = 673 by XPRIMES0:674,675,676,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 458329 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673
  proof
    assume p*p <= k < 458329;
    then p*p < 677*677 by XXREAL_0:2;
    hence thesis by Ttool677a,NAT_4:1;
  end;

theorem Ttool683a:
  p < 683 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677
  proof
    assume p < 683;
    then 1+1 < p+1 & p < 682+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 677;
      hence thesis by Ttool677a;
    end;
    suppose 677 <= p <= 677+1 or 678 <= p <= 678+1 or 679 <= p <= 679+1 or 
      680 <= p <= 680+1 or 681 <= p <= 681+1;
      then p = 677 by XPRIMES0:678,679,680,681,682,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 466489 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677
  proof
    assume p*p <= k < 466489;
    then p*p < 683*683 by XXREAL_0:2;
    hence thesis by Ttool683a,NAT_4:1;
  end;

theorem Ttool691a:
  p < 691 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683
  proof
    assume p < 691;
    then 1+1 < p+1 & p < 690+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 683;
      hence thesis by Ttool683a;
    end;
    suppose 683 <= p <= 683+1 or 684 <= p <= 684+1 or 685 <= p <= 685+1 or 
      686 <= p <= 686+1 or 687 <= p <= 687+1 or 688 <= p <= 688+1 or 
      689 <= p <= 689+1;
      then p = 683 by XPRIMES0:684,685,686,687,688,689,690,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 477481 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683
  proof
    assume p*p <= k < 477481;
    then p*p < 691*691 by XXREAL_0:2;
    hence thesis by Ttool691a,NAT_4:1;
  end;

theorem Ttool701a:
  p < 701 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691
  proof
    assume p < 701;
    then 1+1 < p+1 & p < 700+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 691;
      hence thesis by Ttool691a;
    end;
    suppose 691 <= p <= 691+1 or 692 <= p <= 692+1 or 693 <= p <= 693+1 or 
      694 <= p <= 694+1 or 695 <= p <= 695+1 or 696 <= p <= 696+1 or 
      697 <= p <= 697+1 or 698 <= p <= 698+1 or 699 <= p <= 699+1;
      then p = 691 by XPRIMES0:692,693,694,695,696,697,698,699,700,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 491401 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691
  proof
    assume p*p <= k < 491401;
    then p*p < 701*701 by XXREAL_0:2;
    hence thesis by Ttool701a,NAT_4:1;
  end;

theorem Ttool709a:
  p < 709 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701
  proof
    assume p < 709;
    then 1+1 < p+1 & p < 708+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 701;
      hence thesis by Ttool701a;
    end;
    suppose 701 <= p <= 701+1 or 702 <= p <= 702+1 or 703 <= p <= 703+1 or 
      704 <= p <= 704+1 or 705 <= p <= 705+1 or 706 <= p <= 706+1 or 
      707 <= p <= 707+1;
      then p = 701 by XPRIMES0:702,703,704,705,706,707,708,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 502681 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701
  proof
    assume p*p <= k < 502681;
    then p*p < 709*709 by XXREAL_0:2;
    hence thesis by Ttool709a,NAT_4:1;
  end;

theorem Ttool719a:
  p < 719 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709
  proof
    assume p < 719;
    then 1+1 < p+1 & p < 718+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 709;
      hence thesis by Ttool709a;
    end;
    suppose 709 <= p <= 709+1 or 710 <= p <= 710+1 or 711 <= p <= 711+1 or 
      712 <= p <= 712+1 or 713 <= p <= 713+1 or 714 <= p <= 714+1 or 
      715 <= p <= 715+1 or 716 <= p <= 716+1 or 717 <= p <= 717+1;
      then p = 709 by XPRIMES0:710,711,712,713,714,715,716,717,718,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 516961 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709
  proof
    assume p*p <= k < 516961;
    then p*p < 719*719 by XXREAL_0:2;
    hence thesis by Ttool719a,NAT_4:1;
  end;

theorem Ttool727a:
  p < 727 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719
  proof
    assume p < 727;
    then 1+1 < p+1 & p < 726+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 719;
      hence thesis by Ttool719a;
    end;
    suppose 719 <= p <= 719+1 or 720 <= p <= 720+1 or 721 <= p <= 721+1 or 
      722 <= p <= 722+1 or 723 <= p <= 723+1 or 724 <= p <= 724+1 or 
      725 <= p <= 725+1;
      then p = 719 by XPRIMES0:720,721,722,723,724,725,726,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 528529 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719
  proof
    assume p*p <= k < 528529;
    then p*p < 727*727 by XXREAL_0:2;
    hence thesis by Ttool727a,NAT_4:1;
  end;

theorem Ttool733a:
  p < 733 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727
  proof
    assume p < 733;
    then 1+1 < p+1 & p < 732+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 727;
      hence thesis by Ttool727a;
    end;
    suppose 727 <= p <= 727+1 or 728 <= p <= 728+1 or 729 <= p <= 729+1 or 
      730 <= p <= 730+1 or 731 <= p <= 731+1;
      then p = 727 by XPRIMES0:728,729,730,731,732,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 537289 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727
  proof
    assume p*p <= k < 537289;
    then p*p < 733*733 by XXREAL_0:2;
    hence thesis by Ttool733a,NAT_4:1;
  end;

theorem Ttool739a:
  p < 739 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733
  proof
    assume p < 739;
    then 1+1 < p+1 & p < 738+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 733;
      hence thesis by Ttool733a;
    end;
    suppose 733 <= p <= 733+1 or 734 <= p <= 734+1 or 735 <= p <= 735+1 or 
      736 <= p <= 736+1 or 737 <= p <= 737+1;
      then p = 733 by XPRIMES0:734,735,736,737,738,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 546121 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733
  proof
    assume p*p <= k < 546121;
    then p*p < 739*739 by XXREAL_0:2;
    hence thesis by Ttool739a,NAT_4:1;
  end;

theorem Ttool743a:
  p < 743 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739
  proof
    assume p < 743;
    then 1+1 < p+1 & p < 742+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 739;
      hence thesis by Ttool739a;
    end;
    suppose 739 <= p <= 739+1 or 740 <= p <= 740+1 or 741 <= p <= 741+1;
      then p = 739 by XPRIMES0:740,741,742,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 552049 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739
  proof
    assume p*p <= k < 552049;
    then p*p < 743*743 by XXREAL_0:2;
    hence thesis by Ttool743a,NAT_4:1;
  end;

theorem Ttool751a:
  p < 751 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743
  proof
    assume p < 751;
    then 1+1 < p+1 & p < 750+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 743;
      hence thesis by Ttool743a;
    end;
    suppose 743 <= p <= 743+1 or 744 <= p <= 744+1 or 745 <= p <= 745+1 or 
      746 <= p <= 746+1 or 747 <= p <= 747+1 or 748 <= p <= 748+1 or 
      749 <= p <= 749+1;
      then p = 743 by XPRIMES0:744,745,746,747,748,749,750,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 564001 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743
  proof
    assume p*p <= k < 564001;
    then p*p < 751*751 by XXREAL_0:2;
    hence thesis by Ttool751a,NAT_4:1;
  end;

theorem Ttool757a:
  p < 757 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751
  proof
    assume p < 757;
    then 1+1 < p+1 & p < 756+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 751;
      hence thesis by Ttool751a;
    end;
    suppose 751 <= p <= 751+1 or 752 <= p <= 752+1 or 753 <= p <= 753+1 or 
      754 <= p <= 754+1 or 755 <= p <= 755+1;
      then p = 751 by XPRIMES0:752,753,754,755,756,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 573049 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751
  proof
    assume p*p <= k < 573049;
    then p*p < 757*757 by XXREAL_0:2;
    hence thesis by Ttool757a,NAT_4:1;
  end;

theorem Ttool761a:
  p < 761 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757
  proof
    assume p < 761;
    then 1+1 < p+1 & p < 760+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 757;
      hence thesis by Ttool757a;
    end;
    suppose 757 <= p <= 757+1 or 758 <= p <= 758+1 or 759 <= p <= 759+1;
      then p = 757 by XPRIMES0:758,759,760,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 579121 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757
  proof
    assume p*p <= k < 579121;
    then p*p < 761*761 by XXREAL_0:2;
    hence thesis by Ttool761a,NAT_4:1;
  end;

theorem Ttool769a:
  p < 769 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761
  proof
    assume p < 769;
    then 1+1 < p+1 & p < 768+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 761;
      hence thesis by Ttool761a;
    end;
    suppose 761 <= p <= 761+1 or 762 <= p <= 762+1 or 763 <= p <= 763+1 or 
      764 <= p <= 764+1 or 765 <= p <= 765+1 or 766 <= p <= 766+1 or 
      767 <= p <= 767+1;
      then p = 761 by XPRIMES0:762,763,764,765,766,767,768,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 591361 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761
  proof
    assume p*p <= k < 591361;
    then p*p < 769*769 by XXREAL_0:2;
    hence thesis by Ttool769a,NAT_4:1;
  end;

theorem Ttool773a:
  p < 773 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769
  proof
    assume p < 773;
    then 1+1 < p+1 & p < 772+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 769;
      hence thesis by Ttool769a;
    end;
    suppose 769 <= p <= 769+1 or 770 <= p <= 770+1 or 771 <= p <= 771+1;
      then p = 769 by XPRIMES0:770,771,772,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 597529 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769
  proof
    assume p*p <= k < 597529;
    then p*p < 773*773 by XXREAL_0:2;
    hence thesis by Ttool773a,NAT_4:1;
  end;

theorem Ttool787a:
  p < 787 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773
  proof
    assume p < 787;
    then 1+1 < p+1 & p < 786+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 773;
      hence thesis by Ttool773a;
    end;
    suppose 773 <= p <= 779+1;
      then 773 <= p <= 773+1 or 774 <= p <= 774+1 or 775 <= p <= 775+1 or 
      776 <= p <= 776+1 or 777 <= p <= 777+1 or 778 <= p <= 778+1 or 
      779 <= p <= 779+1;
      then p = 773 by XPRIMES0:774,775,776,777,778,779,780,NAT_1:9;
      hence thesis;
    end;
    suppose 780 <= p <= 780+1 or 781 <= p <= 781+1 or 
      782 <= p <= 782+1 or 783 <= p <= 783+1 or 784 <= p <= 784+1 or 
      785 <= p <= 785+1;
      then contradiction by XPRIMES0:780,781,782,783,784,785,786,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 619369 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773
  proof
    assume p*p <= k < 619369;
    then p*p < 787*787 by XXREAL_0:2;
    hence thesis by Ttool787a,NAT_4:1;
  end;

theorem Ttool797a:
  p < 797 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787
  proof
    assume p < 797;
    then 1+1 < p+1 & p < 796+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 787;
      hence thesis by Ttool787a;
    end;
    suppose 787 <= p <= 787+1 or 788 <= p <= 788+1 or 789 <= p <= 789+1 or 
      790 <= p <= 790+1 or 791 <= p <= 791+1 or 792 <= p <= 792+1 or 
      793 <= p <= 793+1 or 794 <= p <= 794+1 or 795 <= p <= 795+1;
      then p = 787 by XPRIMES0:788,789,790,791,792,793,794,795,796,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 635209 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787
  proof
    assume p*p <= k < 635209;
    then p*p < 797*797 by XXREAL_0:2;
    hence thesis by Ttool797a,NAT_4:1;
  end;

theorem Ttool809a:
  p < 809 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797
  proof
    assume p < 809;
    then 1+1 < p+1 & p < 808+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 797;
      hence thesis by Ttool797a;
    end;
    suppose 797 <= p <= 797+1 or 798 <= p <= 798+1 or 799 <= p <= 799+1 or 
      800 <= p <= 800+1 or 801 <= p <= 801+1 or 802 <= p <= 802+1 or 
      803 <= p <= 803+1 or 804 <= p <= 804+1 or 805 <= p <= 805+1 or 
      806 <= p <= 806+1 or 807 <= p <= 807+1;
      then p = 797 by XPRIMES0:798,799,800,801,802,803,804,805,806,807,808,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 654481 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797
  proof
    assume p*p <= k < 654481;
    then p*p < 809*809 by XXREAL_0:2;
    hence thesis by Ttool809a,NAT_4:1;
  end;

theorem Ttool811a:
  p < 811 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809
  proof
    assume p < 811;
    then 1+1 < p+1 & p < 810+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 809;
      hence thesis by Ttool809a;
    end;
    suppose 809 <= p <= 809+1;
      then p = 809 by XPRIMES0:810,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 657721 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809
  proof
    assume p*p <= k < 657721;
    then p*p < 811*811 by XXREAL_0:2;
    hence thesis by Ttool811a,NAT_4:1;
  end;

theorem Ttool821a:
  p < 821 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811
  proof
    assume p < 821;
    then 1+1 < p+1 & p < 820+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 811;
      hence thesis by Ttool811a;
    end;
    suppose 811 <= p <= 811+1 or 812 <= p <= 812+1 or 813 <= p <= 813+1 or 
      814 <= p <= 814+1 or 815 <= p <= 815+1 or 816 <= p <= 816+1 or 
      817 <= p <= 817+1 or 818 <= p <= 818+1 or 819 <= p <= 819+1;
      then p = 811 by XPRIMES0:812,813,814,815,816,817,818,819,820,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 674041 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811
  proof
    assume p*p <= k < 674041;
    then p*p < 821*821 by XXREAL_0:2;
    hence thesis by Ttool821a,NAT_4:1;
  end;

theorem Ttool823a:
  p < 823 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821
  proof
    assume p < 823;
    then 1+1 < p+1 & p < 822+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 821;
      hence thesis by Ttool821a;
    end;
    suppose 821 <= p <= 821+1;
      then p = 821 by XPRIMES0:822,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 677329 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821
  proof
    assume p*p <= k < 677329;
    then p*p < 823*823 by XXREAL_0:2;
    hence thesis by Ttool823a,NAT_4:1;
  end;

theorem Ttool827a:
  p < 827 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823
  proof
    assume p < 827;
    then 1+1 < p+1 & p < 826+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 823;
      hence thesis by Ttool823a;
    end;
    suppose 823 <= p <= 823+1 or 824 <= p <= 824+1 or 825 <= p <= 825+1;
      then p = 823 by XPRIMES0:824,825,826,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 683929 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823
  proof
    assume p*p <= k < 683929;
    then p*p < 827*827 by XXREAL_0:2;
    hence thesis by Ttool827a,NAT_4:1;
  end;

theorem Ttool829a:
  p < 829 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827
  proof
    assume p < 829;
    then 1+1 < p+1 & p < 828+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 827;
      hence thesis by Ttool827a;
    end;
    suppose 827 <= p <= 827+1;
      then p = 827 by XPRIMES0:828,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 687241 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827
  proof
    assume p*p <= k < 687241;
    then p*p < 829*829 by XXREAL_0:2;
    hence thesis by Ttool829a,NAT_4:1;
  end;

theorem Ttool839a:
  p < 839 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829
  proof
    assume p < 839;
    then 1+1 < p+1 & p < 838+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 829;
      hence thesis by Ttool829a;
    end;
    suppose 829 <= p <= 829+1 or 830 <= p <= 830+1 or 831 <= p <= 831+1 or 
      832 <= p <= 832+1 or 833 <= p <= 833+1 or 834 <= p <= 834+1 or 
      835 <= p <= 835+1 or 836 <= p <= 836+1 or 837 <= p <= 837+1;
      then p = 829 by XPRIMES0:830,831,832,833,834,835,836,837,838,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 703921 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829
  proof
    assume p*p <= k < 703921;
    then p*p < 839*839 by XXREAL_0:2;
    hence thesis by Ttool839a,NAT_4:1;
  end;

theorem Ttool853a:
  p < 853 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839
  proof
    assume p < 853;
    then 1+1 < p+1 & p < 852+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 839;
      hence thesis by Ttool839a;
    end;
    suppose 839 <= p <= 845+1;
      then 839 <= p <= 839+1 or 840 <= p <= 840+1 or 841 <= p <= 841+1 or 
      842 <= p <= 842+1 or 843 <= p <= 843+1 or 844 <= p <= 844+1 or 
      845 <= p <= 845+1;
      then p = 839 by XPRIMES0:840,841,842,843,844,845,846,NAT_1:9;
      hence thesis;
    end;
    suppose 846 <= p <= 846+1 or 847 <= p <= 847+1 or 
      848 <= p <= 848+1 or 849 <= p <= 849+1 or 850 <= p <= 850+1 or 
      851 <= p <= 851+1;
      then contradiction by XPRIMES0:846,847,848,849,850,851,852,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 727609 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839
  proof
    assume p*p <= k < 727609;
    then p*p < 853*853 by XXREAL_0:2;
    hence thesis by Ttool853a,NAT_4:1;
  end;

theorem Ttool857a:
  p < 857 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853
  proof
    assume p < 857;
    then 1+1 < p+1 & p < 856+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 853;
      hence thesis by Ttool853a;
    end;
    suppose 853 <= p <= 853+1 or 854 <= p <= 854+1 or 855 <= p <= 855+1;
      then p = 853 by XPRIMES0:854,855,856,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 734449 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853
  proof
    assume p*p <= k < 734449;
    then p*p < 857*857 by XXREAL_0:2;
    hence thesis by Ttool857a,NAT_4:1;
  end;

theorem Ttool859a:
  p < 859 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857
  proof
    assume p < 859;
    then 1+1 < p+1 & p < 858+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 857;
      hence thesis by Ttool857a;
    end;
    suppose 857 <= p <= 857+1;
      then p = 857 by XPRIMES0:858,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 737881 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857
  proof
    assume p*p <= k < 737881;
    then p*p < 859*859 by XXREAL_0:2;
    hence thesis by Ttool859a,NAT_4:1;
  end;

theorem Ttool863a:
  p < 863 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859
  proof
    assume p < 863;
    then 1+1 < p+1 & p < 862+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 859;
      hence thesis by Ttool859a;
    end;
    suppose 859 <= p <= 859+1 or 860 <= p <= 860+1 or 861 <= p <= 861+1;
      then p = 859 by XPRIMES0:860,861,862,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 744769 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859
  proof
    assume p*p <= k < 744769;
    then p*p < 863*863 by XXREAL_0:2;
    hence thesis by Ttool863a,NAT_4:1;
  end;

theorem Ttool877a:
  p < 877 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863
  proof
    assume p < 877;
    then 1+1 < p+1 & p < 876+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 863;
      hence thesis by Ttool863a;
    end;
    suppose 863 <= p <= 869+1;
      then 863 <= p <= 863+1 or 864 <= p <= 864+1 or 865 <= p <= 865+1 or 
      866 <= p <= 866+1 or 867 <= p <= 867+1 or 868 <= p <= 868+1 or 
      869 <= p <= 869+1;
      then p = 863 by XPRIMES0:864,865,866,867,868,869,870,NAT_1:9;
      hence thesis;
    end;
    suppose 870 <= p <= 870+1 or 871 <= p <= 871+1 or 
      872 <= p <= 872+1 or 873 <= p <= 873+1 or 874 <= p <= 874+1 or 
      875 <= p <= 875+1;
      then contradiction by XPRIMES0:870,871,872,873,874,875,876,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 769129 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863
  proof
    assume p*p <= k < 769129;
    then p*p < 877*877 by XXREAL_0:2;
    hence thesis by Ttool877a,NAT_4:1;
  end;

theorem Ttool881a:
  p < 881 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877
  proof
    assume p < 881;
    then 1+1 < p+1 & p < 880+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 877;
      hence thesis by Ttool877a;
    end;
    suppose 877 <= p <= 877+1 or 878 <= p <= 878+1 or 879 <= p <= 879+1;
      then p = 877 by XPRIMES0:878,879,880,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 776161 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877
  proof
    assume p*p <= k < 776161;
    then p*p < 881*881 by XXREAL_0:2;
    hence thesis by Ttool881a,NAT_4:1;
  end;

theorem Ttool883a:
  p < 883 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881
  proof
    assume p < 883;
    then 1+1 < p+1 & p < 882+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 881;
      hence thesis by Ttool881a;
    end;
    suppose 881 <= p <= 881+1;
      then p = 881 by XPRIMES0:882,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 779689 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881
  proof
    assume p*p <= k < 779689;
    then p*p < 883*883 by XXREAL_0:2;
    hence thesis by Ttool883a,NAT_4:1;
  end;

theorem Ttool887a:
  p < 887 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883
  proof
    assume p < 887;
    then 1+1 < p+1 & p < 886+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 883;
      hence thesis by Ttool883a;
    end;
    suppose 883 <= p <= 883+1 or 884 <= p <= 884+1 or 885 <= p <= 885+1;
      then p = 883 by XPRIMES0:884,885,886,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 786769 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883
  proof
    assume p*p <= k < 786769;
    then p*p < 887*887 by XXREAL_0:2;
    hence thesis by Ttool887a,NAT_4:1;
  end;

theorem Ttool907a:
  p < 907 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887
  proof
    assume p < 907;
    then 1+1 < p+1 & p < 906+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 887;
      hence thesis by Ttool887a;
    end;
    suppose 887 <= p <= 896+1;
      then 887 <= p <= 887+1 or 888 <= p <= 888+1 or 889 <= p <= 889+1 or 
      890 <= p <= 890+1 or 891 <= p <= 891+1 or 892 <= p <= 892+1 or 
      893 <= p <= 893+1 or 894 <= p <= 894+1 or 895 <= p <= 895+1 or 
      896 <= p <= 896+1;
      then p = 887 by XPRIMES0:888,889,890,891,892,893,894,895,896,897,NAT_1:9;
      hence thesis;
    end;
    suppose 897 <= p <= 897+1 or 898 <= p <= 898+1 or 
      899 <= p <= 899+1 or 900 <= p <= 900+1 or 901 <= p <= 901+1 or 
      902 <= p <= 902+1 or 903 <= p <= 903+1 or 904 <= p <= 904+1 or 
      905 <= p <= 905+1;
      then contradiction by XPRIMES0:897,898,899,900,901,902,903,904,905,906,
        NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 822649 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887
  proof
    assume p*p <= k < 822649;
    then p*p < 907*907 by XXREAL_0:2;
    hence thesis by Ttool907a,NAT_4:1;
  end;

theorem Ttool911a:
  p < 911 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907
  proof
    assume p < 911;
    then 1+1 < p+1 & p < 910+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 907;
      hence thesis by Ttool907a;
    end;
    suppose 907 <= p <= 907+1 or 908 <= p <= 908+1 or 909 <= p <= 909+1;
      then p = 907 by XPRIMES0:908,909,910,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 829921 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907
  proof
    assume p*p <= k < 829921;
    then p*p < 911*911 by XXREAL_0:2;
    hence thesis by Ttool911a,NAT_4:1;
  end;

theorem Ttool919a:
  p < 919 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911
  proof
    assume p < 919;
    then 1+1 < p+1 & p < 918+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 911;
      hence thesis by Ttool911a;
    end;
    suppose 911 <= p <= 911+1 or 912 <= p <= 912+1 or 913 <= p <= 913+1 or 
      914 <= p <= 914+1 or 915 <= p <= 915+1 or 916 <= p <= 916+1 or 
      917 <= p <= 917+1;
      then p = 911 by XPRIMES0:912,913,914,915,916,917,918,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 844561 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911
  proof
    assume p*p <= k < 844561;
    then p*p < 919*919 by XXREAL_0:2;
    hence thesis by Ttool919a,NAT_4:1;
  end;

theorem Ttool929a:
  p < 929 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919
  proof
    assume p < 929;
    then 1+1 < p+1 & p < 928+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 919;
      hence thesis by Ttool919a;
    end;
    suppose 919 <= p <= 919+1 or 920 <= p <= 920+1 or 921 <= p <= 921+1 or 
      922 <= p <= 922+1 or 923 <= p <= 923+1 or 924 <= p <= 924+1 or 
      925 <= p <= 925+1 or 926 <= p <= 926+1 or 927 <= p <= 927+1;
      then p = 919 by XPRIMES0:920,921,922,923,924,925,926,927,928,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 863041 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919
  proof
    assume p*p <= k < 863041;
    then p*p < 929*929 by XXREAL_0:2;
    hence thesis by Ttool929a,NAT_4:1;
  end;

theorem Ttool937a:
  p < 937 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929
  proof
    assume p < 937;
    then 1+1 < p+1 & p < 936+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 929;
      hence thesis by Ttool929a;
    end;
    suppose 929 <= p <= 929+1 or 930 <= p <= 930+1 or 931 <= p <= 931+1 or 
      932 <= p <= 932+1 or 933 <= p <= 933+1 or 934 <= p <= 934+1 or 
      935 <= p <= 935+1;
      then p = 929 by XPRIMES0:930,931,932,933,934,935,936,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 877969 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929
  proof
    assume p*p <= k < 877969;
    then p*p < 937*937 by XXREAL_0:2;
    hence thesis by Ttool937a,NAT_4:1;
  end;

theorem Ttool941a:
  p < 941 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937
  proof
    assume p < 941;
    then 1+1 < p+1 & p < 940+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 937;
      hence thesis by Ttool937a;
    end;
    suppose 937 <= p <= 937+1 or 938 <= p <= 938+1 or 939 <= p <= 939+1;
      then p = 937 by XPRIMES0:938,939,940,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 885481 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937
  proof
    assume p*p <= k < 885481;
    then p*p < 941*941 by XXREAL_0:2;
    hence thesis by Ttool941a,NAT_4:1;
  end;

theorem Ttool947a:
  p < 947 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941
  proof
    assume p < 947;
    then 1+1 < p+1 & p < 946+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 941;
      hence thesis by Ttool941a;
    end;
    suppose 941 <= p <= 941+1 or 942 <= p <= 942+1 or 943 <= p <= 943+1 or 
      944 <= p <= 944+1 or 945 <= p <= 945+1;
      then p = 941 by XPRIMES0:942,943,944,945,946,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 896809 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941
  proof
    assume p*p <= k < 896809;
    then p*p < 947*947 by XXREAL_0:2;
    hence thesis by Ttool947a,NAT_4:1;
  end;

theorem Ttool953a:
  p < 953 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947
  proof
    assume p < 953;
    then 1+1 < p+1 & p < 952+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 947;
      hence thesis by Ttool947a;
    end;
    suppose 947 <= p <= 947+1 or 948 <= p <= 948+1 or 949 <= p <= 949+1 or 
      950 <= p <= 950+1 or 951 <= p <= 951+1;
      then p = 947 by XPRIMES0:948,949,950,951,952,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 908209 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947
  proof
    assume p*p <= k < 908209;
    then p*p < 953*953 by XXREAL_0:2;
    hence thesis by Ttool953a,NAT_4:1;
  end;

theorem Ttool967a:
  p < 967 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953
  proof
    assume p < 967;
    then 1+1 < p+1 & p < 966+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 953;
      hence thesis by Ttool953a;
    end;
    suppose 953 <= p <= 959+1;
      then 953 <= p <= 953+1 or 954 <= p <= 954+1 or 955 <= p <= 955+1 or 
      956 <= p <= 956+1 or 957 <= p <= 957+1 or 958 <= p <= 958+1 or 
      959 <= p <= 959+1;
      then p = 953 by XPRIMES0:954,955,956,957,958,959,960,NAT_1:9;
      hence thesis;
    end;
    suppose 960 <= p <= 960+1 or 961 <= p <= 961+1 or 
      962 <= p <= 962+1 or 963 <= p <= 963+1 or 964 <= p <= 964+1 or 
      965 <= p <= 965+1;
      then contradiction by XPRIMES0:960,961,962,963,964,965,966,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 935089 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953
  proof
    assume p*p <= k < 935089;
    then p*p < 967*967 by XXREAL_0:2;
    hence thesis by Ttool967a,NAT_4:1;
  end;

theorem Ttool971a:
  p < 971 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967
  proof
    assume p < 971;
    then 1+1 < p+1 & p < 970+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 967;
      hence thesis by Ttool967a;
    end;
    suppose 967 <= p <= 967+1 or 968 <= p <= 968+1 or 969 <= p <= 969+1;
      then p = 967 by XPRIMES0:968,969,970,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 942841 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967
  proof
    assume p*p <= k < 942841;
    then p*p < 971*971 by XXREAL_0:2;
    hence thesis by Ttool971a,NAT_4:1;
  end;

theorem Ttool977a:
  p < 977 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971
  proof
    assume p < 977;
    then 1+1 < p+1 & p < 976+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 971;
      hence thesis by Ttool971a;
    end;
    suppose 971 <= p <= 971+1 or 972 <= p <= 972+1 or 973 <= p <= 973+1 or 
      974 <= p <= 974+1 or 975 <= p <= 975+1;
      then p = 971 by XPRIMES0:972,973,974,975,976,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 954529 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971
  proof
    assume p*p <= k < 954529;
    then p*p < 977*977 by XXREAL_0:2;
    hence thesis by Ttool977a,NAT_4:1;
  end;

theorem Ttool983a:
  p < 983 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977
  proof
    assume p < 983;
    then 1+1 < p+1 & p < 982+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 977;
      hence thesis by Ttool977a;
    end;
    suppose 977 <= p <= 977+1 or 978 <= p <= 978+1 or 979 <= p <= 979+1 or 
      980 <= p <= 980+1 or 981 <= p <= 981+1;
      then p = 977 by XPRIMES0:978,979,980,981,982,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 966289 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977
  proof
    assume p*p <= k < 966289;
    then p*p < 983*983 by XXREAL_0:2;
    hence thesis by Ttool983a,NAT_4:1;
  end;

theorem Ttool991a:
  p < 991 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977 or p = 983
  proof
    assume p < 991;
    then 1+1 < p+1 & p < 990+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 983;
      hence thesis by Ttool983a;
    end;
    suppose 983 <= p <= 983+1 or 984 <= p <= 984+1 or 985 <= p <= 985+1 or 
      986 <= p <= 986+1 or 987 <= p <= 987+1 or 988 <= p <= 988+1 or 
      989 <= p <= 989+1;
      then p = 983 by XPRIMES0:984,985,986,987,988,989,990,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 982081 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977 or p = 983
  proof
    assume p*p <= k < 982081;
    then p*p < 991*991 by XXREAL_0:2;
    hence thesis by Ttool991a,NAT_4:1;
  end;

theorem Ttool997a:
  p < 997 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977 or p = 983 or p = 991
  proof
    assume p < 997;
    then 1+1 < p+1 & p < 996+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 991;
      hence thesis by Ttool991a;
    end;
    suppose 991 <= p <= 991+1 or 992 <= p <= 992+1 or 993 <= p <= 993+1 or 
      994 <= p <= 994+1 or 995 <= p <= 995+1;
      then p = 991 by XPRIMES0:992,993,994,995,996,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 994009 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977 or p = 983 or p = 991
  proof
    assume p*p <= k < 994009;
    then p*p < 997*997 by XXREAL_0:2;
    hence thesis by Ttool997a,NAT_4:1;
  end;

theorem Ttool1009a:
  p < 1009 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977 or p = 983 or p = 991 or p = 997
  proof
    assume p < 1009;
    then 1+1 < p+1 & p < 1008+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 997;
      hence thesis by Ttool997a;
    end;
    suppose 997 <= p <= 997+1 or 998 <= p <= 998+1 or 999 <= p <= 999+1 or 
      1000 <= p <= 1000+1 or 1001 <= p <= 1001+1 or 1002 <= p <= 1002+1 or 
      1003 <= p <= 1003+1 or 1004 <= p <= 1004+1 or 1005 <= p <= 1005+1 or 
      1006 <= p <= 1006+1 or 1007 <= p <= 1007+1;
      then p = 997 by XPRIMES0:1000,1001,1002,1003,1004,1005,1006,1007,1008,
        998,999,NAT_1:9;
      hence thesis;
    end;
  end;

theorem
  p*p <= k < 1018081 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 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or 
  p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or 
  p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or 
  p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or 
  p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or 
  p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or 
  p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or 
  p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or 
  p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or 
  p = 953 or p = 967 or p = 971 or p = 977 or p = 983 or p = 991 or p = 997
  proof
    assume p*p <= k < 1018081;
    then p*p < 1009*1009 by XXREAL_0:2;
    hence thesis by Ttool1009a,NAT_4:1;
  end;