1171 = (2 * 585) + 1 ;
hence not 2 divides 1171 by NAT_4:9; :: thesis:
1171 = (3 * 390) + 1 ;
hence not 3 divides 1171 by NAT_4:9; :: thesis:
1171 = (5 * 234) + 1 ;
hence not 5 divides 1171 by NAT_4:9; :: thesis:
1171 = (7 * 167) + 2 ;
hence not 7 divides 1171 by NAT_4:9; :: thesis:
1171 = (11 * 106) + 5 ;
hence not 11 divides 1171 by NAT_4:9; :: thesis:
1171 = (13 * 90) + 1 ;
hence not 13 divides 1171 by NAT_4:9; :: thesis:
1171 = (17 * 68) + 15 ;
hence not 17 divides 1171 by NAT_4:9; :: thesis:
1171 = (19 * 61) + 12 ;
hence not 19 divides 1171 by NAT_4:9; :: thesis:
1171 = (23 * 50) + 21 ;
hence not 23 divides 1171 by NAT_4:9; :: thesis:
1171 = (29 * 40) + 11 ;
hence not 29 divides 1171 by NAT_4:9; :: thesis:
1171 = (31 * 37) + 24 ;
hence not 31 divides 1171 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1171 & n is prime holds
not n divides 1171 by XPRIMET1:22;
hence 1171 is prime by NAT_4:14; :: thesis: