now :: thesis: ( not 2 divides 1109 & not 3 divides 1109 & not 5 divides 1109 & not 7 divides 1109 & not 11 divides 1109 & not 13 divides 1109 & not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )

1109 = (2 * 554) + 1 ;
hence not 2 divides 1109 by NAT_4:9; :: thesis: ( not 3 divides 1109 & not 5 divides 1109 & not 7 divides 1109 & not 11 divides 1109 & not 13 divides 1109 & not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (3 * 369) + 2 ;
hence not 3 divides 1109 by NAT_4:9; :: thesis: ( not 5 divides 1109 & not 7 divides 1109 & not 11 divides 1109 & not 13 divides 1109 & not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (5 * 221) + 4 ;
hence not 5 divides 1109 by NAT_4:9; :: thesis: ( not 7 divides 1109 & not 11 divides 1109 & not 13 divides 1109 & not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (7 * 158) + 3 ;
hence not 7 divides 1109 by NAT_4:9; :: thesis: ( not 11 divides 1109 & not 13 divides 1109 & not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (11 * 100) + 9 ;
hence not 11 divides 1109 by NAT_4:9; :: thesis: ( not 13 divides 1109 & not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (13 * 85) + 4 ;
hence not 13 divides 1109 by NAT_4:9; :: thesis: ( not 17 divides 1109 & not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (17 * 65) + 4 ;
hence not 17 divides 1109 by NAT_4:9; :: thesis: ( not 19 divides 1109 & not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (19 * 58) + 7 ;
hence not 19 divides 1109 by NAT_4:9; :: thesis: ( not 23 divides 1109 & not 29 divides 1109 & not 31 divides 1109 )
1109 = (23 * 48) + 5 ;
hence not 23 divides 1109 by NAT_4:9; :: thesis: ( not 29 divides 1109 & not 31 divides 1109 )
1109 = (29 * 38) + 7 ;
hence not 29 divides 1109 by NAT_4:9; :: thesis: not 31 divides 1109
1109 = (31 * 35) + 24 ;
hence not 31 divides 1109 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1109 & n is prime holds
not n divides 1109 by XPRIMET1:22;
hence 1109 is prime by NAT_4:14; :: thesis: verum