1549 = (2 * 774) + 1 ;
hence not 2 divides 1549 by NAT_4:9; :: thesis:
1549 = (3 * 516) + 1 ;
hence not 3 divides 1549 by NAT_4:9; :: thesis:
1549 = (5 * 309) + 4 ;
hence not 5 divides 1549 by NAT_4:9; :: thesis:
1549 = (7 * 221) + 2 ;
hence not 7 divides 1549 by NAT_4:9; :: thesis:
1549 = (11 * 140) + 9 ;
hence not 11 divides 1549 by NAT_4:9; :: thesis:
1549 = (13 * 119) + 2 ;
hence not 13 divides 1549 by NAT_4:9; :: thesis:
1549 = (17 * 91) + 2 ;
hence not 17 divides 1549 by NAT_4:9; :: thesis:
1549 = (19 * 81) + 10 ;
hence not 19 divides 1549 by NAT_4:9; :: thesis:
1549 = (23 * 67) + 8 ;
hence not 23 divides 1549 by NAT_4:9; :: thesis:
1549 = (29 * 53) + 12 ;
hence not 29 divides 1549 by NAT_4:9; :: thesis:
1549 = (31 * 49) + 30 ;
hence not 31 divides 1549 by NAT_4:9; :: thesis:
1549 = (37 * 41) + 32 ;
hence not 37 divides 1549 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1549 & n is prime holds
not n divides 1549 by XPRIMET1:24;
hence 1549 is prime by NAT_4:14; :: thesis: