1831 = (2 * 915) + 1 ;
hence not 2 divides 1831 by NAT_4:9; :: thesis:
1831 = (3 * 610) + 1 ;
hence not 3 divides 1831 by NAT_4:9; :: thesis:
1831 = (5 * 366) + 1 ;
hence not 5 divides 1831 by NAT_4:9; :: thesis:
1831 = (7 * 261) + 4 ;
hence not 7 divides 1831 by NAT_4:9; :: thesis:
1831 = (11 * 166) + 5 ;
hence not 11 divides 1831 by NAT_4:9; :: thesis:
1831 = (13 * 140) + 11 ;
hence not 13 divides 1831 by NAT_4:9; :: thesis:
1831 = (17 * 107) + 12 ;
hence not 17 divides 1831 by NAT_4:9; :: thesis:
1831 = (19 * 96) + 7 ;
hence not 19 divides 1831 by NAT_4:9; :: thesis:
1831 = (23 * 79) + 14 ;
hence not 23 divides 1831 by NAT_4:9; :: thesis:
1831 = (29 * 63) + 4 ;
hence not 29 divides 1831 by NAT_4:9; :: thesis:
1831 = (31 * 59) + 2 ;
hence not 31 divides 1831 by NAT_4:9; :: thesis:
1831 = (37 * 49) + 18 ;
hence not 37 divides 1831 by NAT_4:9; :: thesis:
1831 = (41 * 44) + 27 ;
hence not 41 divides 1831 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1831 & n is prime holds
not n divides 1831 by XPRIMET1:26;
hence 1831 is prime by NAT_4:14; :: thesis: