now :: thesis: ( not 2 divides 1249 & not 3 divides 1249 & not 5 divides 1249 & not 7 divides 1249 & not 11 divides 1249 & not 13 divides 1249 & not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )

1249 = (2 * 624) + 1 ;
hence not 2 divides 1249 by NAT_4:9; :: thesis: ( not 3 divides 1249 & not 5 divides 1249 & not 7 divides 1249 & not 11 divides 1249 & not 13 divides 1249 & not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (3 * 416) + 1 ;
hence not 3 divides 1249 by NAT_4:9; :: thesis: ( not 5 divides 1249 & not 7 divides 1249 & not 11 divides 1249 & not 13 divides 1249 & not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (5 * 249) + 4 ;
hence not 5 divides 1249 by NAT_4:9; :: thesis: ( not 7 divides 1249 & not 11 divides 1249 & not 13 divides 1249 & not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (7 * 178) + 3 ;
hence not 7 divides 1249 by NAT_4:9; :: thesis: ( not 11 divides 1249 & not 13 divides 1249 & not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (11 * 113) + 6 ;
hence not 11 divides 1249 by NAT_4:9; :: thesis: ( not 13 divides 1249 & not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (13 * 96) + 1 ;
hence not 13 divides 1249 by NAT_4:9; :: thesis: ( not 17 divides 1249 & not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (17 * 73) + 8 ;
hence not 17 divides 1249 by NAT_4:9; :: thesis: ( not 19 divides 1249 & not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (19 * 65) + 14 ;
hence not 19 divides 1249 by NAT_4:9; :: thesis: ( not 23 divides 1249 & not 29 divides 1249 & not 31 divides 1249 )
1249 = (23 * 54) + 7 ;
hence not 23 divides 1249 by NAT_4:9; :: thesis: ( not 29 divides 1249 & not 31 divides 1249 )
1249 = (29 * 43) + 2 ;
hence not 29 divides 1249 by NAT_4:9; :: thesis: not 31 divides 1249
1249 = (31 * 40) + 9 ;
hence not 31 divides 1249 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1249 & n is prime holds
not n divides 1249 by XPRIMET1:22;
hence 1249 is prime by NAT_4:14; :: thesis: verum