2731 = (2 * 1365) + 1 ;
hence not 2 divides 2731 by NAT_4:9; :: thesis:
2731 = (3 * 910) + 1 ;
hence not 3 divides 2731 by NAT_4:9; :: thesis:
2731 = (5 * 546) + 1 ;
hence not 5 divides 2731 by NAT_4:9; :: thesis:
2731 = (7 * 390) + 1 ;
hence not 7 divides 2731 by NAT_4:9; :: thesis:
2731 = (11 * 248) + 3 ;
hence not 11 divides 2731 by NAT_4:9; :: thesis:
2731 = (13 * 210) + 1 ;
hence not 13 divides 2731 by NAT_4:9; :: thesis:
2731 = (17 * 160) + 11 ;
hence not 17 divides 2731 by NAT_4:9; :: thesis:
2731 = (19 * 143) + 14 ;
hence not 19 divides 2731 by NAT_4:9; :: thesis:
2731 = (23 * 118) + 17 ;
hence not 23 divides 2731 by NAT_4:9; :: thesis:
2731 = (29 * 94) + 5 ;
hence not 29 divides 2731 by NAT_4:9; :: thesis:
2731 = (31 * 88) + 3 ;
hence not 31 divides 2731 by NAT_4:9; :: thesis:
2731 = (37 * 73) + 30 ;
hence not 37 divides 2731 by NAT_4:9; :: thesis:
2731 = (41 * 66) + 25 ;
hence not 41 divides 2731 by NAT_4:9; :: thesis:
2731 = (43 * 63) + 22 ;
hence not 43 divides 2731 by NAT_4:9; :: thesis:
2731 = (47 * 58) + 5 ;
hence not 47 divides 2731 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2731 & n is prime holds
not n divides 2731 by XPRIMET1:30;
hence 2731 is prime by NAT_4:14; :: thesis: