2017 = (2 * 1008) + 1 ;
hence not 2 divides 2017 by NAT_4:9; :: thesis:
2017 = (3 * 672) + 1 ;
hence not 3 divides 2017 by NAT_4:9; :: thesis:
2017 = (5 * 403) + 2 ;
hence not 5 divides 2017 by NAT_4:9; :: thesis:
2017 = (7 * 288) + 1 ;
hence not 7 divides 2017 by NAT_4:9; :: thesis:
2017 = (11 * 183) + 4 ;
hence not 11 divides 2017 by NAT_4:9; :: thesis:
2017 = (13 * 155) + 2 ;
hence not 13 divides 2017 by NAT_4:9; :: thesis:
2017 = (17 * 118) + 11 ;
hence not 17 divides 2017 by NAT_4:9; :: thesis:
2017 = (19 * 106) + 3 ;
hence not 19 divides 2017 by NAT_4:9; :: thesis:
2017 = (23 * 87) + 16 ;
hence not 23 divides 2017 by NAT_4:9; :: thesis:
2017 = (29 * 69) + 16 ;
hence not 29 divides 2017 by NAT_4:9; :: thesis:
2017 = (31 * 65) + 2 ;
hence not 31 divides 2017 by NAT_4:9; :: thesis:
2017 = (37 * 54) + 19 ;
hence not 37 divides 2017 by NAT_4:9; :: thesis:
2017 = (41 * 49) + 8 ;
hence not 41 divides 2017 by NAT_4:9; :: thesis:
2017 = (43 * 46) + 39 ;
hence not 43 divides 2017 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2017 & n is prime holds
not n divides 2017 by XPRIMET1:28;
hence 2017 is prime by NAT_4:14; :: thesis: