2797 = (2 * 1398) + 1 ;
hence not 2 divides 2797 by NAT_4:9; :: thesis:
2797 = (3 * 932) + 1 ;
hence not 3 divides 2797 by NAT_4:9; :: thesis:
2797 = (5 * 559) + 2 ;
hence not 5 divides 2797 by NAT_4:9; :: thesis:
2797 = (7 * 399) + 4 ;
hence not 7 divides 2797 by NAT_4:9; :: thesis:
2797 = (11 * 254) + 3 ;
hence not 11 divides 2797 by NAT_4:9; :: thesis:
2797 = (13 * 215) + 2 ;
hence not 13 divides 2797 by NAT_4:9; :: thesis:
2797 = (17 * 164) + 9 ;
hence not 17 divides 2797 by NAT_4:9; :: thesis:
2797 = (19 * 147) + 4 ;
hence not 19 divides 2797 by NAT_4:9; :: thesis:
2797 = (23 * 121) + 14 ;
hence not 23 divides 2797 by NAT_4:9; :: thesis:
2797 = (29 * 96) + 13 ;
hence not 29 divides 2797 by NAT_4:9; :: thesis:
2797 = (31 * 90) + 7 ;
hence not 31 divides 2797 by NAT_4:9; :: thesis:
2797 = (37 * 75) + 22 ;
hence not 37 divides 2797 by NAT_4:9; :: thesis:
2797 = (41 * 68) + 9 ;
hence not 41 divides 2797 by NAT_4:9; :: thesis:
2797 = (43 * 65) + 2 ;
hence not 43 divides 2797 by NAT_4:9; :: thesis:
2797 = (47 * 59) + 24 ;
hence not 47 divides 2797 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2797 & n is prime holds
not n divides 2797 by XPRIMET1:30;
hence 2797 is prime by NAT_4:14; :: thesis: