1741 = (2 * 870) + 1 ;
hence not 2 divides 1741 by NAT_4:9; :: thesis:
1741 = (3 * 580) + 1 ;
hence not 3 divides 1741 by NAT_4:9; :: thesis:
1741 = (5 * 348) + 1 ;
hence not 5 divides 1741 by NAT_4:9; :: thesis:
1741 = (7 * 248) + 5 ;
hence not 7 divides 1741 by NAT_4:9; :: thesis:
1741 = (11 * 158) + 3 ;
hence not 11 divides 1741 by NAT_4:9; :: thesis:
1741 = (13 * 133) + 12 ;
hence not 13 divides 1741 by NAT_4:9; :: thesis:
1741 = (17 * 102) + 7 ;
hence not 17 divides 1741 by NAT_4:9; :: thesis:
1741 = (19 * 91) + 12 ;
hence not 19 divides 1741 by NAT_4:9; :: thesis:
1741 = (23 * 75) + 16 ;
hence not 23 divides 1741 by NAT_4:9; :: thesis:
1741 = (29 * 60) + 1 ;
hence not 29 divides 1741 by NAT_4:9; :: thesis:
1741 = (31 * 56) + 5 ;
hence not 31 divides 1741 by NAT_4:9; :: thesis:
1741 = (37 * 47) + 2 ;
hence not 37 divides 1741 by NAT_4:9; :: thesis:
1741 = (41 * 42) + 19 ;
hence not 41 divides 1741 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1741 & n is prime holds
not n divides 1741 by XPRIMET1:26;
hence 1741 is prime by NAT_4:14; :: thesis: