4801 = (2 * 2400) + 1 ;
hence not 2 divides 4801 by NAT_4:9; :: thesis:
4801 = (3 * 1600) + 1 ;
hence not 3 divides 4801 by NAT_4:9; :: thesis:
4801 = (5 * 960) + 1 ;
hence not 5 divides 4801 by NAT_4:9; :: thesis:
4801 = (7 * 685) + 6 ;
hence not 7 divides 4801 by NAT_4:9; :: thesis:
4801 = (11 * 436) + 5 ;
hence not 11 divides 4801 by NAT_4:9; :: thesis:
4801 = (13 * 369) + 4 ;
hence not 13 divides 4801 by NAT_4:9; :: thesis:
4801 = (17 * 282) + 7 ;
hence not 17 divides 4801 by NAT_4:9; :: thesis:
4801 = (19 * 252) + 13 ;
hence not 19 divides 4801 by NAT_4:9; :: thesis:
4801 = (23 * 208) + 17 ;
hence not 23 divides 4801 by NAT_4:9; :: thesis:
4801 = (29 * 165) + 16 ;
hence not 29 divides 4801 by NAT_4:9; :: thesis:
4801 = (31 * 154) + 27 ;
hence not 31 divides 4801 by NAT_4:9; :: thesis:
4801 = (37 * 129) + 28 ;
hence not 37 divides 4801 by NAT_4:9; :: thesis:
4801 = (41 * 117) + 4 ;
hence not 41 divides 4801 by NAT_4:9; :: thesis:
4801 = (43 * 111) + 28 ;
hence not 43 divides 4801 by NAT_4:9; :: thesis:
4801 = (47 * 102) + 7 ;
hence not 47 divides 4801 by NAT_4:9; :: thesis:
4801 = (53 * 90) + 31 ;
hence not 53 divides 4801 by NAT_4:9; :: thesis:
4801 = (59 * 81) + 22 ;
hence not 59 divides 4801 by NAT_4:9; :: thesis:
4801 = (61 * 78) + 43 ;
hence not 61 divides 4801 by NAT_4:9; :: thesis:
4801 = (67 * 71) + 44 ;
hence not 67 divides 4801 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4801 & n is prime holds
not n divides 4801 by XPRIMET1:38;
hence 4801 is prime by NAT_4:14; :: thesis: