4523 = (2 * 2261) + 1 ;
hence not 2 divides 4523 by NAT_4:9; :: thesis:
4523 = (3 * 1507) + 2 ;
hence not 3 divides 4523 by NAT_4:9; :: thesis:
4523 = (5 * 904) + 3 ;
hence not 5 divides 4523 by NAT_4:9; :: thesis:
4523 = (7 * 646) + 1 ;
hence not 7 divides 4523 by NAT_4:9; :: thesis:
4523 = (11 * 411) + 2 ;
hence not 11 divides 4523 by NAT_4:9; :: thesis:
4523 = (13 * 347) + 12 ;
hence not 13 divides 4523 by NAT_4:9; :: thesis:
4523 = (17 * 266) + 1 ;
hence not 17 divides 4523 by NAT_4:9; :: thesis:
4523 = (19 * 238) + 1 ;
hence not 19 divides 4523 by NAT_4:9; :: thesis:
4523 = (23 * 196) + 15 ;
hence not 23 divides 4523 by NAT_4:9; :: thesis:
4523 = (29 * 155) + 28 ;
hence not 29 divides 4523 by NAT_4:9; :: thesis:
4523 = (31 * 145) + 28 ;
hence not 31 divides 4523 by NAT_4:9; :: thesis:
4523 = (37 * 122) + 9 ;
hence not 37 divides 4523 by NAT_4:9; :: thesis:
4523 = (41 * 110) + 13 ;
hence not 41 divides 4523 by NAT_4:9; :: thesis:
4523 = (43 * 105) + 8 ;
hence not 43 divides 4523 by NAT_4:9; :: thesis:
4523 = (47 * 96) + 11 ;
hence not 47 divides 4523 by NAT_4:9; :: thesis:
4523 = (53 * 85) + 18 ;
hence not 53 divides 4523 by NAT_4:9; :: thesis:
4523 = (59 * 76) + 39 ;
hence not 59 divides 4523 by NAT_4:9; :: thesis:
4523 = (61 * 74) + 9 ;
hence not 61 divides 4523 by NAT_4:9; :: thesis:
4523 = (67 * 67) + 34 ;
hence not 67 divides 4523 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4523 & n is prime holds
not n divides 4523 by XPRIMET1:38;
hence 4523 is prime by NAT_4:14; :: thesis: