1291 = (2 * 645) + 1 ;
hence not 2 divides 1291 by NAT_4:9; :: thesis:
1291 = (3 * 430) + 1 ;
hence not 3 divides 1291 by NAT_4:9; :: thesis:
1291 = (5 * 258) + 1 ;
hence not 5 divides 1291 by NAT_4:9; :: thesis:
1291 = (7 * 184) + 3 ;
hence not 7 divides 1291 by NAT_4:9; :: thesis:
1291 = (11 * 117) + 4 ;
hence not 11 divides 1291 by NAT_4:9; :: thesis:
1291 = (13 * 99) + 4 ;
hence not 13 divides 1291 by NAT_4:9; :: thesis:
1291 = (17 * 75) + 16 ;
hence not 17 divides 1291 by NAT_4:9; :: thesis:
1291 = (19 * 67) + 18 ;
hence not 19 divides 1291 by NAT_4:9; :: thesis:
1291 = (23 * 56) + 3 ;
hence not 23 divides 1291 by NAT_4:9; :: thesis:
1291 = (29 * 44) + 15 ;
hence not 29 divides 1291 by NAT_4:9; :: thesis:
1291 = (31 * 41) + 20 ;
hence not 31 divides 1291 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1291 & n is prime holds
not n divides 1291 by XPRIMET1:22;
hence 1291 is prime by NAT_4:14; :: thesis: