now :: thesis: ( not 2 divides 443 & not 3 divides 443 & not 5 divides 443 & not 7 divides 443 & not 11 divides 443 & not 13 divides 443 & not 17 divides 443 & not 19 divides 443 )

443 = (2 * 221) + 1 ;
hence not 2 divides 443 by NAT_4:9; :: thesis: ( not 3 divides 443 & not 5 divides 443 & not 7 divides 443 & not 11 divides 443 & not 13 divides 443 & not 17 divides 443 & not 19 divides 443 )
443 = (3 * 147) + 2 ;
hence not 3 divides 443 by NAT_4:9; :: thesis: ( not 5 divides 443 & not 7 divides 443 & not 11 divides 443 & not 13 divides 443 & not 17 divides 443 & not 19 divides 443 )
443 = (5 * 88) + 3 ;
hence not 5 divides 443 by NAT_4:9; :: thesis: ( not 7 divides 443 & not 11 divides 443 & not 13 divides 443 & not 17 divides 443 & not 19 divides 443 )
443 = (7 * 63) + 2 ;
hence not 7 divides 443 by NAT_4:9; :: thesis: ( not 11 divides 443 & not 13 divides 443 & not 17 divides 443 & not 19 divides 443 )
443 = (11 * 40) + 3 ;
hence not 11 divides 443 by NAT_4:9; :: thesis: ( not 13 divides 443 & not 17 divides 443 & not 19 divides 443 )
443 = (13 * 34) + 1 ;
hence not 13 divides 443 by NAT_4:9; :: thesis: ( not 17 divides 443 & not 19 divides 443 )
443 = (17 * 26) + 1 ;
hence not 17 divides 443 by NAT_4:9; :: thesis: not 19 divides 443
443 = (19 * 23) + 6 ;
hence not 19 divides 443 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 443 & n is prime holds
not n divides 443 by XPRIMET1:16;
hence 443 is prime by NAT_4:14; :: thesis: verum