33331 = (2 * 16665) + 1 ;
hence not 2 divides 33331 ; :: thesis:
33331 = (3 * 11110) + 1 ;
hence not 3 divides 33331 by NAT_4:9; :: thesis:
33331 = (5 * 6666) + 1 ;
hence not 5 divides 33331 by NAT_4:9; :: thesis:
33331 = (7 * 4761) + 4 ;
hence not 7 divides 33331 by NAT_4:9; :: thesis:
33331 = (11 * 3030) + 1 ;
hence not 11 divides 33331 by NAT_4:9; :: thesis:
33331 = (13 * 2563) + 12 ;
hence not 13 divides 33331 by NAT_4:9; :: thesis:
33331 = (17 * 1960) + 11 ;
hence not 17 divides 33331 by NAT_4:9; :: thesis:
33331 = (19 * 1754) + 5 ;
hence not 19 divides 33331 by NAT_4:9; :: thesis:
33331 = (23 * 1449) + 4 ;
hence not 23 divides 33331 by NAT_4:9; :: thesis:
33331 = (29 * 1149) + 10 ;
hence not 29 divides 33331 by NAT_4:9; :: thesis:
33331 = (31 * 1075) + 6 ;
hence not 31 divides 33331 by NAT_4:9; :: thesis:
33331 = (37 * 900) + 31 ;
hence not 37 divides 33331 by NAT_4:9; :: thesis:
33331 = (41 * 812) + 39 ;
hence not 41 divides 33331 by NAT_4:9; :: thesis:
33331 = (43 * 775) + 6 ;
hence not 43 divides 33331 by NAT_4:9; :: thesis:
33331 = (47 * 709) + 8 ;
hence not 47 divides 33331 by NAT_4:9; :: thesis:
33331 = (53 * 628) + 47 ;
hence not 53 divides 33331 by NAT_4:9; :: thesis:
33331 = (59 * 564) + 55 ;
hence not 59 divides 33331 by NAT_4:9; :: thesis:
33331 = (61 * 546) + 25 ;
hence not 61 divides 33331 by NAT_4:9; :: thesis:
33331 = (67 * 497) + 32 ;
hence not 67 divides 33331 by NAT_4:9; :: thesis:
33331 = (71 * 469) + 32 ;
hence not 71 divides 33331 by NAT_4:9; :: thesis:
33331 = (73 * 456) + 43 ;
hence not 73 divides 33331 by NAT_4:9; :: thesis:
33331 = (79 * 421) + 72 ;
hence not 79 divides 33331 by NAT_4:9; :: thesis:
33331 = (83 * 401) + 48 ;
hence not 83 divides 33331 by NAT_4:9; :: thesis:
33331 = (89 * 374) + 45 ;
hence not 89 divides 33331 by NAT_4:9; :: thesis:
33331 = (97 * 343) + 60 ;
hence not 97 divides 33331 by NAT_4:9; :: thesis:
33331 = (101 * 330) + 1 ;
hence not 101 divides 33331 by NAT_4:9; :: thesis:
33331 = (101 * 330) + 1 ;
hence not 101 divides 33331 by NAT_4:9; :: thesis:
33331 = (103 * 323) + 62 ;
hence not 103 divides 33331 by NAT_4:9; :: thesis:
33331 = (107 * 311) + 54 ;
hence not 107 divides 33331 by NAT_4:9; :: thesis:
33331 = (109 * 305) + 86 ;
hence not 109 divides 33331 by NAT_4:9; :: thesis:
33331 = (113 * 294) + 109 ;
hence not 113 divides 33331 by NAT_4:9; :: thesis:
33331 = (127 * 262) + 57 ;
hence not 127 divides 33331 by NAT_4:9; :: thesis:
33331 = (131 * 254) + 57 ;
hence not 131 divides 33331 by NAT_4:9; :: thesis:
33331 = (137 * 243) + 40 ;
hence not 137 divides 33331 by NAT_4:9; :: thesis:
33331 = (139 * 239) + 110 ;
hence not 139 divides 33331 by NAT_4:9; :: thesis:
33331 = (149 * 223) + 104 ;
hence not 149 divides 33331 by NAT_4:9; :: thesis:
33331 = (151 * 220) + 111 ;
hence not 151 divides 33331 by NAT_4:9; :: thesis:
33331 = (157 * 212) + 47 ;
hence not 157 divides 33331 by NAT_4:9; :: thesis:
33331 = (163 * 204) + 79 ;
hence not 163 divides 33331 by NAT_4:9; :: thesis:
33331 = (167 * 199) + 98 ;
hence not 167 divides 33331 by NAT_4:9; :: thesis:
33331 = (173 * 192) + 115 ;
hence not 173 divides 33331 by NAT_4:9; :: thesis:
33331 = (179 * 186) + 37 ;
hence not 179 divides 33331 by NAT_4:9; :: thesis:
33331 = (181 * 184) + 27 ;
hence not 181 divides 33331 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 33331 & n is prime holds
not n divides 33331 by XPRIMET1:84;
hence 33331 is prime by NAT_4:14; :: thesis: