1723 = (2 * 861) + 1 ;
hence not 2 divides 1723 by NAT_4:9; :: thesis:
1723 = (3 * 574) + 1 ;
hence not 3 divides 1723 by NAT_4:9; :: thesis:
1723 = (5 * 344) + 3 ;
hence not 5 divides 1723 by NAT_4:9; :: thesis:
1723 = (7 * 246) + 1 ;
hence not 7 divides 1723 by NAT_4:9; :: thesis:
1723 = (11 * 156) + 7 ;
hence not 11 divides 1723 by NAT_4:9; :: thesis:
1723 = (13 * 132) + 7 ;
hence not 13 divides 1723 by NAT_4:9; :: thesis:
1723 = (17 * 101) + 6 ;
hence not 17 divides 1723 by NAT_4:9; :: thesis:
1723 = (19 * 90) + 13 ;
hence not 19 divides 1723 by NAT_4:9; :: thesis:
1723 = (23 * 74) + 21 ;
hence not 23 divides 1723 by NAT_4:9; :: thesis:
1723 = (29 * 59) + 12 ;
hence not 29 divides 1723 by NAT_4:9; :: thesis:
1723 = (31 * 55) + 18 ;
hence not 31 divides 1723 by NAT_4:9; :: thesis:
1723 = (37 * 46) + 21 ;
hence not 37 divides 1723 by NAT_4:9; :: thesis:
1723 = (41 * 42) + 1 ;
hence not 41 divides 1723 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1723 & n is prime holds
not n divides 1723 by XPRIMET1:26;
hence 1723 is prime by NAT_4:14; :: thesis: