5749 = (2 * 2874) + 1 ;
hence not 2 divides 5749 by NAT_4:9; :: thesis:
5749 = (3 * 1916) + 1 ;
hence not 3 divides 5749 by NAT_4:9; :: thesis:
5749 = (5 * 1149) + 4 ;
hence not 5 divides 5749 by NAT_4:9; :: thesis:
5749 = (7 * 821) + 2 ;
hence not 7 divides 5749 by NAT_4:9; :: thesis:
5749 = (11 * 522) + 7 ;
hence not 11 divides 5749 by NAT_4:9; :: thesis:
5749 = (13 * 442) + 3 ;
hence not 13 divides 5749 by NAT_4:9; :: thesis:
5749 = (17 * 338) + 3 ;
hence not 17 divides 5749 by NAT_4:9; :: thesis:
5749 = (19 * 302) + 11 ;
hence not 19 divides 5749 by NAT_4:9; :: thesis:
5749 = (23 * 249) + 22 ;
hence not 23 divides 5749 by NAT_4:9; :: thesis:
5749 = (29 * 198) + 7 ;
hence not 29 divides 5749 by NAT_4:9; :: thesis:
5749 = (31 * 185) + 14 ;
hence not 31 divides 5749 by NAT_4:9; :: thesis:
5749 = (37 * 155) + 14 ;
hence not 37 divides 5749 by NAT_4:9; :: thesis:
5749 = (41 * 140) + 9 ;
hence not 41 divides 5749 by NAT_4:9; :: thesis:
5749 = (43 * 133) + 30 ;
hence not 43 divides 5749 by NAT_4:9; :: thesis:
5749 = (47 * 122) + 15 ;
hence not 47 divides 5749 by NAT_4:9; :: thesis:
5749 = (53 * 108) + 25 ;
hence not 53 divides 5749 by NAT_4:9; :: thesis:
5749 = (59 * 97) + 26 ;
hence not 59 divides 5749 by NAT_4:9; :: thesis:
5749 = (61 * 94) + 15 ;
hence not 61 divides 5749 by NAT_4:9; :: thesis:
5749 = (67 * 85) + 54 ;
hence not 67 divides 5749 by NAT_4:9; :: thesis:
5749 = (71 * 80) + 69 ;
hence not 71 divides 5749 by NAT_4:9; :: thesis:
5749 = (73 * 78) + 55 ;
hence not 73 divides 5749 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5749 & n is prime holds
not n divides 5749 by XPRIMET1:42;
hence 5749 is prime by NAT_4:14; :: thesis: