1091 = (2 * 545) + 1 ;
hence not 2 divides 1091 by NAT_4:9; :: thesis:
1091 = (3 * 363) + 2 ;
hence not 3 divides 1091 by NAT_4:9; :: thesis:
1091 = (5 * 218) + 1 ;
hence not 5 divides 1091 by NAT_4:9; :: thesis:
1091 = (7 * 155) + 6 ;
hence not 7 divides 1091 by NAT_4:9; :: thesis:
1091 = (11 * 99) + 2 ;
hence not 11 divides 1091 by NAT_4:9; :: thesis:
1091 = (13 * 83) + 12 ;
hence not 13 divides 1091 by NAT_4:9; :: thesis:
1091 = (17 * 64) + 3 ;
hence not 17 divides 1091 by NAT_4:9; :: thesis:
1091 = (19 * 57) + 8 ;
hence not 19 divides 1091 by NAT_4:9; :: thesis:
1091 = (23 * 47) + 10 ;
hence not 23 divides 1091 by NAT_4:9; :: thesis:
1091 = (29 * 37) + 18 ;
hence not 29 divides 1091 by NAT_4:9; :: thesis:
1091 = (31 * 35) + 6 ;
hence not 31 divides 1091 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1091 & n is prime holds
not n divides 1091 by XPRIMET1:22;
hence 1091 is prime by NAT_4:14; :: thesis: