2707 = (2 * 1353) + 1 ;
hence not 2 divides 2707 by NAT_4:9; :: thesis:
2707 = (3 * 902) + 1 ;
hence not 3 divides 2707 by NAT_4:9; :: thesis:
2707 = (5 * 541) + 2 ;
hence not 5 divides 2707 by NAT_4:9; :: thesis:
2707 = (7 * 386) + 5 ;
hence not 7 divides 2707 by NAT_4:9; :: thesis:
2707 = (11 * 246) + 1 ;
hence not 11 divides 2707 by NAT_4:9; :: thesis:
2707 = (13 * 208) + 3 ;
hence not 13 divides 2707 by NAT_4:9; :: thesis:
2707 = (17 * 159) + 4 ;
hence not 17 divides 2707 by NAT_4:9; :: thesis:
2707 = (19 * 142) + 9 ;
hence not 19 divides 2707 by NAT_4:9; :: thesis:
2707 = (23 * 117) + 16 ;
hence not 23 divides 2707 by NAT_4:9; :: thesis:
2707 = (29 * 93) + 10 ;
hence not 29 divides 2707 by NAT_4:9; :: thesis:
2707 = (31 * 87) + 10 ;
hence not 31 divides 2707 by NAT_4:9; :: thesis:
2707 = (37 * 73) + 6 ;
hence not 37 divides 2707 by NAT_4:9; :: thesis:
2707 = (41 * 66) + 1 ;
hence not 41 divides 2707 by NAT_4:9; :: thesis:
2707 = (43 * 62) + 41 ;
hence not 43 divides 2707 by NAT_4:9; :: thesis:
2707 = (47 * 57) + 28 ;
hence not 47 divides 2707 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2707 & n is prime holds
not n divides 2707 by XPRIMET1:30;
hence 2707 is prime by NAT_4:14; :: thesis: