4621 = (2 * 2310) + 1 ;
hence not 2 divides 4621 by NAT_4:9; :: thesis:
4621 = (3 * 1540) + 1 ;
hence not 3 divides 4621 by NAT_4:9; :: thesis:
4621 = (5 * 924) + 1 ;
hence not 5 divides 4621 by NAT_4:9; :: thesis:
4621 = (7 * 660) + 1 ;
hence not 7 divides 4621 by NAT_4:9; :: thesis:
4621 = (11 * 420) + 1 ;
hence not 11 divides 4621 by NAT_4:9; :: thesis:
4621 = (13 * 355) + 6 ;
hence not 13 divides 4621 by NAT_4:9; :: thesis:
4621 = (17 * 271) + 14 ;
hence not 17 divides 4621 by NAT_4:9; :: thesis:
4621 = (19 * 243) + 4 ;
hence not 19 divides 4621 by NAT_4:9; :: thesis:
4621 = (23 * 200) + 21 ;
hence not 23 divides 4621 by NAT_4:9; :: thesis:
4621 = (29 * 159) + 10 ;
hence not 29 divides 4621 by NAT_4:9; :: thesis:
4621 = (31 * 149) + 2 ;
hence not 31 divides 4621 by NAT_4:9; :: thesis:
4621 = (37 * 124) + 33 ;
hence not 37 divides 4621 by NAT_4:9; :: thesis:
4621 = (41 * 112) + 29 ;
hence not 41 divides 4621 by NAT_4:9; :: thesis:
4621 = (43 * 107) + 20 ;
hence not 43 divides 4621 by NAT_4:9; :: thesis:
4621 = (47 * 98) + 15 ;
hence not 47 divides 4621 by NAT_4:9; :: thesis:
4621 = (53 * 87) + 10 ;
hence not 53 divides 4621 by NAT_4:9; :: thesis:
4621 = (59 * 78) + 19 ;
hence not 59 divides 4621 by NAT_4:9; :: thesis:
4621 = (61 * 75) + 46 ;
hence not 61 divides 4621 by NAT_4:9; :: thesis:
4621 = (67 * 68) + 65 ;
hence not 67 divides 4621 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4621 & n is prime holds
not n divides 4621 by XPRIMET1:38;
hence 4621 is prime by NAT_4:14; :: thesis: