7867 = (2 * 3933) + 1 ;
hence not 2 divides 7867 by NAT_4:9; :: thesis:
7867 = (3 * 2622) + 1 ;
hence not 3 divides 7867 by NAT_4:9; :: thesis:
7867 = (5 * 1573) + 2 ;
hence not 5 divides 7867 by NAT_4:9; :: thesis:
7867 = (7 * 1123) + 6 ;
hence not 7 divides 7867 by NAT_4:9; :: thesis:
7867 = (11 * 715) + 2 ;
hence not 11 divides 7867 by NAT_4:9; :: thesis:
7867 = (13 * 605) + 2 ;
hence not 13 divides 7867 by NAT_4:9; :: thesis:
7867 = (17 * 462) + 13 ;
hence not 17 divides 7867 by NAT_4:9; :: thesis:
7867 = (19 * 414) + 1 ;
hence not 19 divides 7867 by NAT_4:9; :: thesis:
7867 = (23 * 342) + 1 ;
hence not 23 divides 7867 by NAT_4:9; :: thesis:
7867 = (29 * 271) + 8 ;
hence not 29 divides 7867 by NAT_4:9; :: thesis:
7867 = (31 * 253) + 24 ;
hence not 31 divides 7867 by NAT_4:9; :: thesis:
7867 = (37 * 212) + 23 ;
hence not 37 divides 7867 by NAT_4:9; :: thesis:
7867 = (41 * 191) + 36 ;
hence not 41 divides 7867 by NAT_4:9; :: thesis:
7867 = (43 * 182) + 41 ;
hence not 43 divides 7867 by NAT_4:9; :: thesis:
7867 = (47 * 167) + 18 ;
hence not 47 divides 7867 by NAT_4:9; :: thesis:
7867 = (53 * 148) + 23 ;
hence not 53 divides 7867 by NAT_4:9; :: thesis:
7867 = (59 * 133) + 20 ;
hence not 59 divides 7867 by NAT_4:9; :: thesis:
7867 = (61 * 128) + 59 ;
hence not 61 divides 7867 by NAT_4:9; :: thesis:
7867 = (67 * 117) + 28 ;
hence not 67 divides 7867 by NAT_4:9; :: thesis:
7867 = (71 * 110) + 57 ;
hence not 71 divides 7867 by NAT_4:9; :: thesis:
7867 = (73 * 107) + 56 ;
hence not 73 divides 7867 by NAT_4:9; :: thesis:
7867 = (79 * 99) + 46 ;
hence not 79 divides 7867 by NAT_4:9; :: thesis:
7867 = (83 * 94) + 65 ;
hence not 83 divides 7867 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7867 & n is prime holds
not n divides 7867 by XPRIMET1:46;
hence 7867 is prime by NAT_4:14; :: thesis: