4493 = (2 * 2246) + 1 ;
hence not 2 divides 4493 by NAT_4:9; :: thesis:
4493 = (3 * 1497) + 2 ;
hence not 3 divides 4493 by NAT_4:9; :: thesis:
4493 = (5 * 898) + 3 ;
hence not 5 divides 4493 by NAT_4:9; :: thesis:
4493 = (7 * 641) + 6 ;
hence not 7 divides 4493 by NAT_4:9; :: thesis:
4493 = (11 * 408) + 5 ;
hence not 11 divides 4493 by NAT_4:9; :: thesis:
4493 = (13 * 345) + 8 ;
hence not 13 divides 4493 by NAT_4:9; :: thesis:
4493 = (17 * 264) + 5 ;
hence not 17 divides 4493 by NAT_4:9; :: thesis:
4493 = (19 * 236) + 9 ;
hence not 19 divides 4493 by NAT_4:9; :: thesis:
4493 = (23 * 195) + 8 ;
hence not 23 divides 4493 by NAT_4:9; :: thesis:
4493 = (29 * 154) + 27 ;
hence not 29 divides 4493 by NAT_4:9; :: thesis:
4493 = (31 * 144) + 29 ;
hence not 31 divides 4493 by NAT_4:9; :: thesis:
4493 = (37 * 121) + 16 ;
hence not 37 divides 4493 by NAT_4:9; :: thesis:
4493 = (41 * 109) + 24 ;
hence not 41 divides 4493 by NAT_4:9; :: thesis:
4493 = (43 * 104) + 21 ;
hence not 43 divides 4493 by NAT_4:9; :: thesis:
4493 = (47 * 95) + 28 ;
hence not 47 divides 4493 by NAT_4:9; :: thesis:
4493 = (53 * 84) + 41 ;
hence not 53 divides 4493 by NAT_4:9; :: thesis:
4493 = (59 * 76) + 9 ;
hence not 59 divides 4493 by NAT_4:9; :: thesis:
4493 = (61 * 73) + 40 ;
hence not 61 divides 4493 by NAT_4:9; :: thesis:
4493 = (67 * 67) + 4 ;
hence not 67 divides 4493 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4493 & n is prime holds
not n divides 4493 by XPRIMET1:38;
hence 4493 is prime by NAT_4:14; :: thesis: