1049 = (2 * 524) + 1 ;
hence not 2 divides 1049 by NAT_4:9; :: thesis:
1049 = (3 * 349) + 2 ;
hence not 3 divides 1049 by NAT_4:9; :: thesis:
1049 = (5 * 209) + 4 ;
hence not 5 divides 1049 by NAT_4:9; :: thesis:
1049 = (7 * 149) + 6 ;
hence not 7 divides 1049 by NAT_4:9; :: thesis:
1049 = (11 * 95) + 4 ;
hence not 11 divides 1049 by NAT_4:9; :: thesis:
1049 = (13 * 80) + 9 ;
hence not 13 divides 1049 by NAT_4:9; :: thesis:
1049 = (17 * 61) + 12 ;
hence not 17 divides 1049 by NAT_4:9; :: thesis:
1049 = (19 * 55) + 4 ;
hence not 19 divides 1049 by NAT_4:9; :: thesis:
1049 = (23 * 45) + 14 ;
hence not 23 divides 1049 by NAT_4:9; :: thesis:
1049 = (29 * 36) + 5 ;
hence not 29 divides 1049 by NAT_4:9; :: thesis:
1049 = (31 * 33) + 26 ;
hence not 31 divides 1049 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1049 & n is prime holds
not n divides 1049 by XPRIMET1:22;
hence 1049 is prime by NAT_4:14; :: thesis: