now :: thesis: ( not 2 divides 283 & not 3 divides 283 & not 5 divides 283 & not 7 divides 283 & not 11 divides 283 & not 13 divides 283 )

283 = (2 * 141) + 1 ;
hence not 2 divides 283 by NAT_4:9; :: thesis: ( not 3 divides 283 & not 5 divides 283 & not 7 divides 283 & not 11 divides 283 & not 13 divides 283 )
283 = (3 * 94) + 1 ;
hence not 3 divides 283 by NAT_4:9; :: thesis: ( not 5 divides 283 & not 7 divides 283 & not 11 divides 283 & not 13 divides 283 )
283 = (5 * 56) + 3 ;
hence not 5 divides 283 by NAT_4:9; :: thesis: ( not 7 divides 283 & not 11 divides 283 & not 13 divides 283 )
283 = (7 * 40) + 3 ;
hence not 7 divides 283 by NAT_4:9; :: thesis: ( not 11 divides 283 & not 13 divides 283 )
283 = (11 * 25) + 8 ;
hence not 11 divides 283 by NAT_4:9; :: thesis: not 13 divides 283
283 = (13 * 21) + 10 ;
hence not 13 divides 283 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 283 & n is prime holds
not n divides 283 by XPRIMET1:12;
hence 283 is prime by NAT_4:14; :: thesis: verum