977 = (2 * 488) + 1 ;
hence not 2 divides 977 by NAT_4:9; :: thesis:
977 = (3 * 325) + 2 ;
hence not 3 divides 977 by NAT_4:9; :: thesis:
977 = (5 * 195) + 2 ;
hence not 5 divides 977 by NAT_4:9; :: thesis:
977 = (7 * 139) + 4 ;
hence not 7 divides 977 by NAT_4:9; :: thesis:
977 = (11 * 88) + 9 ;
hence not 11 divides 977 by NAT_4:9; :: thesis:
977 = (13 * 75) + 2 ;
hence not 13 divides 977 by NAT_4:9; :: thesis:
977 = (17 * 57) + 8 ;
hence not 17 divides 977 by NAT_4:9; :: thesis:
977 = (19 * 51) + 8 ;
hence not 19 divides 977 by NAT_4:9; :: thesis:
977 = (23 * 42) + 11 ;
hence not 23 divides 977 by NAT_4:9; :: thesis:
977 = (29 * 33) + 20 ;
hence not 29 divides 977 by NAT_4:9; :: thesis:
977 = (31 * 31) + 16 ;
hence not 31 divides 977 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 977 & n is prime holds
not n divides 977 by XPRIMET1:22;
hence 977 is prime by NAT_4:14; :: thesis: