4603 = (2 * 2301) + 1 ;
hence not 2 divides 4603 by NAT_4:9; :: thesis:
4603 = (3 * 1534) + 1 ;
hence not 3 divides 4603 by NAT_4:9; :: thesis:
4603 = (5 * 920) + 3 ;
hence not 5 divides 4603 by NAT_4:9; :: thesis:
4603 = (7 * 657) + 4 ;
hence not 7 divides 4603 by NAT_4:9; :: thesis:
4603 = (11 * 418) + 5 ;
hence not 11 divides 4603 by NAT_4:9; :: thesis:
4603 = (13 * 354) + 1 ;
hence not 13 divides 4603 by NAT_4:9; :: thesis:
4603 = (17 * 270) + 13 ;
hence not 17 divides 4603 by NAT_4:9; :: thesis:
4603 = (19 * 242) + 5 ;
hence not 19 divides 4603 by NAT_4:9; :: thesis:
4603 = (23 * 200) + 3 ;
hence not 23 divides 4603 by NAT_4:9; :: thesis:
4603 = (29 * 158) + 21 ;
hence not 29 divides 4603 by NAT_4:9; :: thesis:
4603 = (31 * 148) + 15 ;
hence not 31 divides 4603 by NAT_4:9; :: thesis:
4603 = (37 * 124) + 15 ;
hence not 37 divides 4603 by NAT_4:9; :: thesis:
4603 = (41 * 112) + 11 ;
hence not 41 divides 4603 by NAT_4:9; :: thesis:
4603 = (43 * 107) + 2 ;
hence not 43 divides 4603 by NAT_4:9; :: thesis:
4603 = (47 * 97) + 44 ;
hence not 47 divides 4603 by NAT_4:9; :: thesis:
4603 = (53 * 86) + 45 ;
hence not 53 divides 4603 by NAT_4:9; :: thesis:
4603 = (59 * 78) + 1 ;
hence not 59 divides 4603 by NAT_4:9; :: thesis:
4603 = (61 * 75) + 28 ;
hence not 61 divides 4603 by NAT_4:9; :: thesis:
4603 = (67 * 68) + 47 ;
hence not 67 divides 4603 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4603 & n is prime holds
not n divides 4603 by XPRIMET1:38;
hence 4603 is prime by NAT_4:14; :: thesis: