now :: thesis: ( not 2 divides 463 & not 3 divides 463 & not 5 divides 463 & not 7 divides 463 & not 11 divides 463 & not 13 divides 463 & not 17 divides 463 & not 19 divides 463 )

463 = (2 * 231) + 1 ;
hence not 2 divides 463 by NAT_4:9; :: thesis: ( not 3 divides 463 & not 5 divides 463 & not 7 divides 463 & not 11 divides 463 & not 13 divides 463 & not 17 divides 463 & not 19 divides 463 )
463 = (3 * 154) + 1 ;
hence not 3 divides 463 by NAT_4:9; :: thesis: ( not 5 divides 463 & not 7 divides 463 & not 11 divides 463 & not 13 divides 463 & not 17 divides 463 & not 19 divides 463 )
463 = (5 * 92) + 3 ;
hence not 5 divides 463 by NAT_4:9; :: thesis: ( not 7 divides 463 & not 11 divides 463 & not 13 divides 463 & not 17 divides 463 & not 19 divides 463 )
463 = (7 * 66) + 1 ;
hence not 7 divides 463 by NAT_4:9; :: thesis: ( not 11 divides 463 & not 13 divides 463 & not 17 divides 463 & not 19 divides 463 )
463 = (11 * 42) + 1 ;
hence not 11 divides 463 by NAT_4:9; :: thesis: ( not 13 divides 463 & not 17 divides 463 & not 19 divides 463 )
463 = (13 * 35) + 8 ;
hence not 13 divides 463 by NAT_4:9; :: thesis: ( not 17 divides 463 & not 19 divides 463 )
463 = (17 * 27) + 4 ;
hence not 17 divides 463 by NAT_4:9; :: thesis: not 19 divides 463
463 = (19 * 24) + 7 ;
hence not 19 divides 463 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 463 & n is prime holds
not n divides 463 by XPRIMET1:16;
hence 463 is prime by NAT_4:14; :: thesis: verum