now :: thesis: ( not 2 divides 1093 & not 3 divides 1093 & not 5 divides 1093 & not 7 divides 1093 & not 11 divides 1093 & not 13 divides 1093 & not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )

1093 = (2 * 546) + 1 ;
hence not 2 divides 1093 by NAT_4:9; :: thesis: ( not 3 divides 1093 & not 5 divides 1093 & not 7 divides 1093 & not 11 divides 1093 & not 13 divides 1093 & not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (3 * 364) + 1 ;
hence not 3 divides 1093 by NAT_4:9; :: thesis: ( not 5 divides 1093 & not 7 divides 1093 & not 11 divides 1093 & not 13 divides 1093 & not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (5 * 218) + 3 ;
hence not 5 divides 1093 by NAT_4:9; :: thesis: ( not 7 divides 1093 & not 11 divides 1093 & not 13 divides 1093 & not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (7 * 156) + 1 ;
hence not 7 divides 1093 by NAT_4:9; :: thesis: ( not 11 divides 1093 & not 13 divides 1093 & not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (11 * 99) + 4 ;
hence not 11 divides 1093 by NAT_4:9; :: thesis: ( not 13 divides 1093 & not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (13 * 84) + 1 ;
hence not 13 divides 1093 by NAT_4:9; :: thesis: ( not 17 divides 1093 & not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (17 * 64) + 5 ;
hence not 17 divides 1093 by NAT_4:9; :: thesis: ( not 19 divides 1093 & not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (19 * 57) + 10 ;
hence not 19 divides 1093 by NAT_4:9; :: thesis: ( not 23 divides 1093 & not 29 divides 1093 & not 31 divides 1093 )
1093 = (23 * 47) + 12 ;
hence not 23 divides 1093 by NAT_4:9; :: thesis: ( not 29 divides 1093 & not 31 divides 1093 )
1093 = (29 * 37) + 20 ;
hence not 29 divides 1093 by NAT_4:9; :: thesis: not 31 divides 1093
1093 = (31 * 35) + 8 ;
hence not 31 divides 1093 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1093 & n is prime holds
not n divides 1093 by XPRIMET1:22;
hence 1093 is prime by NAT_4:14; :: thesis: verum