5003 = (2 * 2501) + 1 ;
hence not 2 divides 5003 by NAT_4:9; :: thesis:
5003 = (3 * 1667) + 2 ;
hence not 3 divides 5003 by NAT_4:9; :: thesis:
5003 = (5 * 1000) + 3 ;
hence not 5 divides 5003 by NAT_4:9; :: thesis:
5003 = (7 * 714) + 5 ;
hence not 7 divides 5003 by NAT_4:9; :: thesis:
5003 = (11 * 454) + 9 ;
hence not 11 divides 5003 by NAT_4:9; :: thesis:
5003 = (13 * 384) + 11 ;
hence not 13 divides 5003 by NAT_4:9; :: thesis:
5003 = (17 * 294) + 5 ;
hence not 17 divides 5003 by NAT_4:9; :: thesis:
5003 = (19 * 263) + 6 ;
hence not 19 divides 5003 by NAT_4:9; :: thesis:
5003 = (23 * 217) + 12 ;
hence not 23 divides 5003 by NAT_4:9; :: thesis:
5003 = (29 * 172) + 15 ;
hence not 29 divides 5003 by NAT_4:9; :: thesis:
5003 = (31 * 161) + 12 ;
hence not 31 divides 5003 by NAT_4:9; :: thesis:
5003 = (37 * 135) + 8 ;
hence not 37 divides 5003 by NAT_4:9; :: thesis:
5003 = (41 * 122) + 1 ;
hence not 41 divides 5003 by NAT_4:9; :: thesis:
5003 = (43 * 116) + 15 ;
hence not 43 divides 5003 by NAT_4:9; :: thesis:
5003 = (47 * 106) + 21 ;
hence not 47 divides 5003 by NAT_4:9; :: thesis:
5003 = (53 * 94) + 21 ;
hence not 53 divides 5003 by NAT_4:9; :: thesis:
5003 = (59 * 84) + 47 ;
hence not 59 divides 5003 by NAT_4:9; :: thesis:
5003 = (61 * 82) + 1 ;
hence not 61 divides 5003 by NAT_4:9; :: thesis:
5003 = (67 * 74) + 45 ;
hence not 67 divides 5003 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5003 & n is prime holds
not n divides 5003 by XPRIMET1:38;
hence 5003 is prime by NAT_4:14; :: thesis: