now :: thesis: ( not 2 divides 421 & not 3 divides 421 & not 5 divides 421 & not 7 divides 421 & not 11 divides 421 & not 13 divides 421 & not 17 divides 421 & not 19 divides 421 )

421 = (2 * 210) + 1 ;
hence not 2 divides 421 by NAT_4:9; :: thesis: ( not 3 divides 421 & not 5 divides 421 & not 7 divides 421 & not 11 divides 421 & not 13 divides 421 & not 17 divides 421 & not 19 divides 421 )
421 = (3 * 140) + 1 ;
hence not 3 divides 421 by NAT_4:9; :: thesis: ( not 5 divides 421 & not 7 divides 421 & not 11 divides 421 & not 13 divides 421 & not 17 divides 421 & not 19 divides 421 )
421 = (5 * 84) + 1 ;
hence not 5 divides 421 by NAT_4:9; :: thesis: ( not 7 divides 421 & not 11 divides 421 & not 13 divides 421 & not 17 divides 421 & not 19 divides 421 )
421 = (7 * 60) + 1 ;
hence not 7 divides 421 by NAT_4:9; :: thesis: ( not 11 divides 421 & not 13 divides 421 & not 17 divides 421 & not 19 divides 421 )
421 = (11 * 38) + 3 ;
hence not 11 divides 421 by NAT_4:9; :: thesis: ( not 13 divides 421 & not 17 divides 421 & not 19 divides 421 )
421 = (13 * 32) + 5 ;
hence not 13 divides 421 by NAT_4:9; :: thesis: ( not 17 divides 421 & not 19 divides 421 )
421 = (17 * 24) + 13 ;
hence not 17 divides 421 by NAT_4:9; :: thesis: not 19 divides 421
421 = (19 * 22) + 3 ;
hence not 19 divides 421 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 421 & n is prime holds
not n divides 421 by XPRIMET1:16;
hence 421 is prime by NAT_4:14; :: thesis: verum