4373 = (2 * 2186) + 1 ;
hence not 2 divides 4373 by NAT_4:9; :: thesis:
4373 = (3 * 1457) + 2 ;
hence not 3 divides 4373 by NAT_4:9; :: thesis:
4373 = (5 * 874) + 3 ;
hence not 5 divides 4373 by NAT_4:9; :: thesis:
4373 = (7 * 624) + 5 ;
hence not 7 divides 4373 by NAT_4:9; :: thesis:
4373 = (11 * 397) + 6 ;
hence not 11 divides 4373 by NAT_4:9; :: thesis:
4373 = (13 * 336) + 5 ;
hence not 13 divides 4373 by NAT_4:9; :: thesis:
4373 = (17 * 257) + 4 ;
hence not 17 divides 4373 by NAT_4:9; :: thesis:
4373 = (19 * 230) + 3 ;
hence not 19 divides 4373 by NAT_4:9; :: thesis:
4373 = (23 * 190) + 3 ;
hence not 23 divides 4373 by NAT_4:9; :: thesis:
4373 = (29 * 150) + 23 ;
hence not 29 divides 4373 by NAT_4:9; :: thesis:
4373 = (31 * 141) + 2 ;
hence not 31 divides 4373 by NAT_4:9; :: thesis:
4373 = (37 * 118) + 7 ;
hence not 37 divides 4373 by NAT_4:9; :: thesis:
4373 = (41 * 106) + 27 ;
hence not 41 divides 4373 by NAT_4:9; :: thesis:
4373 = (43 * 101) + 30 ;
hence not 43 divides 4373 by NAT_4:9; :: thesis:
4373 = (47 * 93) + 2 ;
hence not 47 divides 4373 by NAT_4:9; :: thesis:
4373 = (53 * 82) + 27 ;
hence not 53 divides 4373 by NAT_4:9; :: thesis:
4373 = (59 * 74) + 7 ;
hence not 59 divides 4373 by NAT_4:9; :: thesis:
4373 = (61 * 71) + 42 ;
hence not 61 divides 4373 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4373 & n is prime holds
not n divides 4373 by XPRIMET1:36;
hence 4373 is prime by NAT_4:14; :: thesis: