4567 = (2 * 2283) + 1 ;
hence not 2 divides 4567 by NAT_4:9; :: thesis:
4567 = (3 * 1522) + 1 ;
hence not 3 divides 4567 by NAT_4:9; :: thesis:
4567 = (5 * 913) + 2 ;
hence not 5 divides 4567 by NAT_4:9; :: thesis:
4567 = (7 * 652) + 3 ;
hence not 7 divides 4567 by NAT_4:9; :: thesis:
4567 = (11 * 415) + 2 ;
hence not 11 divides 4567 by NAT_4:9; :: thesis:
4567 = (13 * 351) + 4 ;
hence not 13 divides 4567 by NAT_4:9; :: thesis:
4567 = (17 * 268) + 11 ;
hence not 17 divides 4567 by NAT_4:9; :: thesis:
4567 = (19 * 240) + 7 ;
hence not 19 divides 4567 by NAT_4:9; :: thesis:
4567 = (23 * 198) + 13 ;
hence not 23 divides 4567 by NAT_4:9; :: thesis:
4567 = (29 * 157) + 14 ;
hence not 29 divides 4567 by NAT_4:9; :: thesis:
4567 = (31 * 147) + 10 ;
hence not 31 divides 4567 by NAT_4:9; :: thesis:
4567 = (37 * 123) + 16 ;
hence not 37 divides 4567 by NAT_4:9; :: thesis:
4567 = (41 * 111) + 16 ;
hence not 41 divides 4567 by NAT_4:9; :: thesis:
4567 = (43 * 106) + 9 ;
hence not 43 divides 4567 by NAT_4:9; :: thesis:
4567 = (47 * 97) + 8 ;
hence not 47 divides 4567 by NAT_4:9; :: thesis:
4567 = (53 * 86) + 9 ;
hence not 53 divides 4567 by NAT_4:9; :: thesis:
4567 = (59 * 77) + 24 ;
hence not 59 divides 4567 by NAT_4:9; :: thesis:
4567 = (61 * 74) + 53 ;
hence not 61 divides 4567 by NAT_4:9; :: thesis:
4567 = (67 * 68) + 11 ;
hence not 67 divides 4567 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4567 & n is prime holds
not n divides 4567 by XPRIMET1:38;
hence 4567 is prime by NAT_4:14; :: thesis: