now :: thesis: ( not 2 divides 839 & not 3 divides 839 & not 5 divides 839 & not 7 divides 839 & not 11 divides 839 & not 13 divides 839 & not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )

839 = (2 * 419) + 1 ;
hence not 2 divides 839 by NAT_4:9; :: thesis: ( not 3 divides 839 & not 5 divides 839 & not 7 divides 839 & not 11 divides 839 & not 13 divides 839 & not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )
839 = (3 * 279) + 2 ;
hence not 3 divides 839 by NAT_4:9; :: thesis: ( not 5 divides 839 & not 7 divides 839 & not 11 divides 839 & not 13 divides 839 & not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )
839 = (5 * 167) + 4 ;
hence not 5 divides 839 by NAT_4:9; :: thesis: ( not 7 divides 839 & not 11 divides 839 & not 13 divides 839 & not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )
839 = (7 * 119) + 6 ;
hence not 7 divides 839 by NAT_4:9; :: thesis: ( not 11 divides 839 & not 13 divides 839 & not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )
839 = (11 * 76) + 3 ;
hence not 11 divides 839 by NAT_4:9; :: thesis: ( not 13 divides 839 & not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )
839 = (13 * 64) + 7 ;
hence not 13 divides 839 by NAT_4:9; :: thesis: ( not 17 divides 839 & not 19 divides 839 & not 23 divides 839 )
839 = (17 * 49) + 6 ;
hence not 17 divides 839 by NAT_4:9; :: thesis: ( not 19 divides 839 & not 23 divides 839 )
839 = (19 * 44) + 3 ;
hence not 19 divides 839 by NAT_4:9; :: thesis: not 23 divides 839
839 = (23 * 36) + 11 ;
hence not 23 divides 839 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 839 & n is prime holds
not n divides 839 by XPRIMET1:18;
hence 839 is prime by NAT_4:14; :: thesis: verum