5023 = (2 * 2511) + 1 ;
hence not 2 divides 5023 by NAT_4:9; :: thesis:
5023 = (3 * 1674) + 1 ;
hence not 3 divides 5023 by NAT_4:9; :: thesis:
5023 = (5 * 1004) + 3 ;
hence not 5 divides 5023 by NAT_4:9; :: thesis:
5023 = (7 * 717) + 4 ;
hence not 7 divides 5023 by NAT_4:9; :: thesis:
5023 = (11 * 456) + 7 ;
hence not 11 divides 5023 by NAT_4:9; :: thesis:
5023 = (13 * 386) + 5 ;
hence not 13 divides 5023 by NAT_4:9; :: thesis:
5023 = (17 * 295) + 8 ;
hence not 17 divides 5023 by NAT_4:9; :: thesis:
5023 = (19 * 264) + 7 ;
hence not 19 divides 5023 by NAT_4:9; :: thesis:
5023 = (23 * 218) + 9 ;
hence not 23 divides 5023 by NAT_4:9; :: thesis:
5023 = (29 * 173) + 6 ;
hence not 29 divides 5023 by NAT_4:9; :: thesis:
5023 = (31 * 162) + 1 ;
hence not 31 divides 5023 by NAT_4:9; :: thesis:
5023 = (37 * 135) + 28 ;
hence not 37 divides 5023 by NAT_4:9; :: thesis:
5023 = (41 * 122) + 21 ;
hence not 41 divides 5023 by NAT_4:9; :: thesis:
5023 = (43 * 116) + 35 ;
hence not 43 divides 5023 by NAT_4:9; :: thesis:
5023 = (47 * 106) + 41 ;
hence not 47 divides 5023 by NAT_4:9; :: thesis:
5023 = (53 * 94) + 41 ;
hence not 53 divides 5023 by NAT_4:9; :: thesis:
5023 = (59 * 85) + 8 ;
hence not 59 divides 5023 by NAT_4:9; :: thesis:
5023 = (61 * 82) + 21 ;
hence not 61 divides 5023 by NAT_4:9; :: thesis:
5023 = (67 * 74) + 65 ;
hence not 67 divides 5023 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5023 & n is prime holds
not n divides 5023 by XPRIMET1:38;
hence 5023 is prime by NAT_4:14; :: thesis: