991 = (2 * 495) + 1 ;
hence not 2 divides 991 by NAT_4:9; :: thesis:
991 = (3 * 330) + 1 ;
hence not 3 divides 991 by NAT_4:9; :: thesis:
991 = (5 * 198) + 1 ;
hence not 5 divides 991 by NAT_4:9; :: thesis:
991 = (7 * 141) + 4 ;
hence not 7 divides 991 by NAT_4:9; :: thesis:
991 = (11 * 90) + 1 ;
hence not 11 divides 991 by NAT_4:9; :: thesis:
991 = (13 * 76) + 3 ;
hence not 13 divides 991 by NAT_4:9; :: thesis:
991 = (17 * 58) + 5 ;
hence not 17 divides 991 by NAT_4:9; :: thesis:
991 = (19 * 52) + 3 ;
hence not 19 divides 991 by NAT_4:9; :: thesis:
991 = (23 * 43) + 2 ;
hence not 23 divides 991 by NAT_4:9; :: thesis:
991 = (29 * 34) + 5 ;
hence not 29 divides 991 by NAT_4:9; :: thesis:
991 = (31 * 31) + 30 ;
hence not 31 divides 991 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 991 & n is prime holds
not n divides 991 by XPRIMET1:22;
hence 991 is prime by NAT_4:14; :: thesis: