4679 = (2 * 2339) + 1 ;
hence not 2 divides 4679 by NAT_4:9; :: thesis:
4679 = (3 * 1559) + 2 ;
hence not 3 divides 4679 by NAT_4:9; :: thesis:
4679 = (5 * 935) + 4 ;
hence not 5 divides 4679 by NAT_4:9; :: thesis:
4679 = (7 * 668) + 3 ;
hence not 7 divides 4679 by NAT_4:9; :: thesis:
4679 = (11 * 425) + 4 ;
hence not 11 divides 4679 by NAT_4:9; :: thesis:
4679 = (13 * 359) + 12 ;
hence not 13 divides 4679 by NAT_4:9; :: thesis:
4679 = (17 * 275) + 4 ;
hence not 17 divides 4679 by NAT_4:9; :: thesis:
4679 = (19 * 246) + 5 ;
hence not 19 divides 4679 by NAT_4:9; :: thesis:
4679 = (23 * 203) + 10 ;
hence not 23 divides 4679 by NAT_4:9; :: thesis:
4679 = (29 * 161) + 10 ;
hence not 29 divides 4679 by NAT_4:9; :: thesis:
4679 = (31 * 150) + 29 ;
hence not 31 divides 4679 by NAT_4:9; :: thesis:
4679 = (37 * 126) + 17 ;
hence not 37 divides 4679 by NAT_4:9; :: thesis:
4679 = (41 * 114) + 5 ;
hence not 41 divides 4679 by NAT_4:9; :: thesis:
4679 = (43 * 108) + 35 ;
hence not 43 divides 4679 by NAT_4:9; :: thesis:
4679 = (47 * 99) + 26 ;
hence not 47 divides 4679 by NAT_4:9; :: thesis:
4679 = (53 * 88) + 15 ;
hence not 53 divides 4679 by NAT_4:9; :: thesis:
4679 = (59 * 79) + 18 ;
hence not 59 divides 4679 by NAT_4:9; :: thesis:
4679 = (61 * 76) + 43 ;
hence not 61 divides 4679 by NAT_4:9; :: thesis:
4679 = (67 * 69) + 56 ;
hence not 67 divides 4679 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4679 & n is prime holds
not n divides 4679 by XPRIMET1:38;
hence 4679 is prime by NAT_4:14; :: thesis: