now :: thesis: ( not 2 divides 1303 & not 3 divides 1303 & not 5 divides 1303 & not 7 divides 1303 & not 11 divides 1303 & not 13 divides 1303 & not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )

1303 = (2 * 651) + 1 ;
hence not 2 divides 1303 by NAT_4:9; :: thesis: ( not 3 divides 1303 & not 5 divides 1303 & not 7 divides 1303 & not 11 divides 1303 & not 13 divides 1303 & not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (3 * 434) + 1 ;
hence not 3 divides 1303 by NAT_4:9; :: thesis: ( not 5 divides 1303 & not 7 divides 1303 & not 11 divides 1303 & not 13 divides 1303 & not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (5 * 260) + 3 ;
hence not 5 divides 1303 by NAT_4:9; :: thesis: ( not 7 divides 1303 & not 11 divides 1303 & not 13 divides 1303 & not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (7 * 186) + 1 ;
hence not 7 divides 1303 by NAT_4:9; :: thesis: ( not 11 divides 1303 & not 13 divides 1303 & not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (11 * 118) + 5 ;
hence not 11 divides 1303 by NAT_4:9; :: thesis: ( not 13 divides 1303 & not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (13 * 100) + 3 ;
hence not 13 divides 1303 by NAT_4:9; :: thesis: ( not 17 divides 1303 & not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (17 * 76) + 11 ;
hence not 17 divides 1303 by NAT_4:9; :: thesis: ( not 19 divides 1303 & not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (19 * 68) + 11 ;
hence not 19 divides 1303 by NAT_4:9; :: thesis: ( not 23 divides 1303 & not 29 divides 1303 & not 31 divides 1303 )
1303 = (23 * 56) + 15 ;
hence not 23 divides 1303 by NAT_4:9; :: thesis: ( not 29 divides 1303 & not 31 divides 1303 )
1303 = (29 * 44) + 27 ;
hence not 29 divides 1303 by NAT_4:9; :: thesis: not 31 divides 1303
1303 = (31 * 42) + 1 ;
hence not 31 divides 1303 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1303 & n is prime holds
not n divides 1303 by XPRIMET1:22;
hence 1303 is prime by NAT_4:14; :: thesis: verum