1721 = (2 * 860) + 1 ;
hence not 2 divides 1721 by NAT_4:9; :: thesis:
1721 = (3 * 573) + 2 ;
hence not 3 divides 1721 by NAT_4:9; :: thesis:
1721 = (5 * 344) + 1 ;
hence not 5 divides 1721 by NAT_4:9; :: thesis:
1721 = (7 * 245) + 6 ;
hence not 7 divides 1721 by NAT_4:9; :: thesis:
1721 = (11 * 156) + 5 ;
hence not 11 divides 1721 by NAT_4:9; :: thesis:
1721 = (13 * 132) + 5 ;
hence not 13 divides 1721 by NAT_4:9; :: thesis:
1721 = (17 * 101) + 4 ;
hence not 17 divides 1721 by NAT_4:9; :: thesis:
1721 = (19 * 90) + 11 ;
hence not 19 divides 1721 by NAT_4:9; :: thesis:
1721 = (23 * 74) + 19 ;
hence not 23 divides 1721 by NAT_4:9; :: thesis:
1721 = (29 * 59) + 10 ;
hence not 29 divides 1721 by NAT_4:9; :: thesis:
1721 = (31 * 55) + 16 ;
hence not 31 divides 1721 by NAT_4:9; :: thesis:
1721 = (37 * 46) + 19 ;
hence not 37 divides 1721 by NAT_4:9; :: thesis:
1721 = (41 * 41) + 40 ;
hence not 41 divides 1721 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1721 & n is prime holds
not n divides 1721 by XPRIMET1:26;
hence 1721 is prime by NAT_4:14; :: thesis: