1231 = (2 * 615) + 1 ;
hence not 2 divides 1231 by NAT_4:9; :: thesis:
1231 = (3 * 410) + 1 ;
hence not 3 divides 1231 by NAT_4:9; :: thesis:
1231 = (5 * 246) + 1 ;
hence not 5 divides 1231 by NAT_4:9; :: thesis:
1231 = (7 * 175) + 6 ;
hence not 7 divides 1231 by NAT_4:9; :: thesis:
1231 = (11 * 111) + 10 ;
hence not 11 divides 1231 by NAT_4:9; :: thesis:
1231 = (13 * 94) + 9 ;
hence not 13 divides 1231 by NAT_4:9; :: thesis:
1231 = (17 * 72) + 7 ;
hence not 17 divides 1231 by NAT_4:9; :: thesis:
1231 = (19 * 64) + 15 ;
hence not 19 divides 1231 by NAT_4:9; :: thesis:
1231 = (23 * 53) + 12 ;
hence not 23 divides 1231 by NAT_4:9; :: thesis:
1231 = (29 * 42) + 13 ;
hence not 29 divides 1231 by NAT_4:9; :: thesis:
1231 = (31 * 39) + 22 ;
hence not 31 divides 1231 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1231 & n is prime holds
not n divides 1231 by XPRIMET1:22;
hence 1231 is prime by NAT_4:14; :: thesis: