971 = (2 * 485) + 1 ;
hence not 2 divides 971 by NAT_4:9; :: thesis:
971 = (3 * 323) + 2 ;
hence not 3 divides 971 by NAT_4:9; :: thesis:
971 = (5 * 194) + 1 ;
hence not 5 divides 971 by NAT_4:9; :: thesis:
971 = (7 * 138) + 5 ;
hence not 7 divides 971 by NAT_4:9; :: thesis:
971 = (11 * 88) + 3 ;
hence not 11 divides 971 by NAT_4:9; :: thesis:
971 = (13 * 74) + 9 ;
hence not 13 divides 971 by NAT_4:9; :: thesis:
971 = (17 * 57) + 2 ;
hence not 17 divides 971 by NAT_4:9; :: thesis:
971 = (19 * 51) + 2 ;
hence not 19 divides 971 by NAT_4:9; :: thesis:
971 = (23 * 42) + 5 ;
hence not 23 divides 971 by NAT_4:9; :: thesis:
971 = (29 * 33) + 14 ;
hence not 29 divides 971 by NAT_4:9; :: thesis:
971 = (31 * 31) + 10 ;
hence not 31 divides 971 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 971 & n is prime holds
not n divides 971 by XPRIMET1:22;
hence 971 is prime by NAT_4:14; :: thesis: