3023 = (2 * 1511) + 1 ;
hence not 2 divides 3023 by NAT_4:9; :: thesis:
3023 = (3 * 1007) + 2 ;
hence not 3 divides 3023 by NAT_4:9; :: thesis:
3023 = (5 * 604) + 3 ;
hence not 5 divides 3023 by NAT_4:9; :: thesis:
3023 = (7 * 431) + 6 ;
hence not 7 divides 3023 by NAT_4:9; :: thesis:
3023 = (11 * 274) + 9 ;
hence not 11 divides 3023 by NAT_4:9; :: thesis:
3023 = (13 * 232) + 7 ;
hence not 13 divides 3023 by NAT_4:9; :: thesis:
3023 = (17 * 177) + 14 ;
hence not 17 divides 3023 by NAT_4:9; :: thesis:
3023 = (19 * 159) + 2 ;
hence not 19 divides 3023 by NAT_4:9; :: thesis:
3023 = (23 * 131) + 10 ;
hence not 23 divides 3023 by NAT_4:9; :: thesis:
3023 = (29 * 104) + 7 ;
hence not 29 divides 3023 by NAT_4:9; :: thesis:
3023 = (31 * 97) + 16 ;
hence not 31 divides 3023 by NAT_4:9; :: thesis:
3023 = (37 * 81) + 26 ;
hence not 37 divides 3023 by NAT_4:9; :: thesis:
3023 = (41 * 73) + 30 ;
hence not 41 divides 3023 by NAT_4:9; :: thesis:
3023 = (43 * 70) + 13 ;
hence not 43 divides 3023 by NAT_4:9; :: thesis:
3023 = (47 * 64) + 15 ;
hence not 47 divides 3023 by NAT_4:9; :: thesis:
3023 = (53 * 57) + 2 ;
hence not 53 divides 3023 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3023 & n is prime holds
not n divides 3023 by XPRIMET1:32;
hence 3023 is prime by NAT_4:14; :: thesis: