2729 = (2 * 1364) + 1 ;
hence not 2 divides 2729 by NAT_4:9; :: thesis:
2729 = (3 * 909) + 2 ;
hence not 3 divides 2729 by NAT_4:9; :: thesis:
2729 = (5 * 545) + 4 ;
hence not 5 divides 2729 by NAT_4:9; :: thesis:
2729 = (7 * 389) + 6 ;
hence not 7 divides 2729 by NAT_4:9; :: thesis:
2729 = (11 * 248) + 1 ;
hence not 11 divides 2729 by NAT_4:9; :: thesis:
2729 = (13 * 209) + 12 ;
hence not 13 divides 2729 by NAT_4:9; :: thesis:
2729 = (17 * 160) + 9 ;
hence not 17 divides 2729 by NAT_4:9; :: thesis:
2729 = (19 * 143) + 12 ;
hence not 19 divides 2729 by NAT_4:9; :: thesis:
2729 = (23 * 118) + 15 ;
hence not 23 divides 2729 by NAT_4:9; :: thesis:
2729 = (29 * 94) + 3 ;
hence not 29 divides 2729 by NAT_4:9; :: thesis:
2729 = (31 * 88) + 1 ;
hence not 31 divides 2729 by NAT_4:9; :: thesis:
2729 = (37 * 73) + 28 ;
hence not 37 divides 2729 by NAT_4:9; :: thesis:
2729 = (41 * 66) + 23 ;
hence not 41 divides 2729 by NAT_4:9; :: thesis:
2729 = (43 * 63) + 20 ;
hence not 43 divides 2729 by NAT_4:9; :: thesis:
2729 = (47 * 58) + 3 ;
hence not 47 divides 2729 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2729 & n is prime holds
not n divides 2729 by XPRIMET1:30;
hence 2729 is prime by NAT_4:14; :: thesis: