now :: thesis: ( not 2 divides 709 & not 3 divides 709 & not 5 divides 709 & not 7 divides 709 & not 11 divides 709 & not 13 divides 709 & not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )

709 = (2 * 354) + 1 ;
hence not 2 divides 709 by NAT_4:9; :: thesis: ( not 3 divides 709 & not 5 divides 709 & not 7 divides 709 & not 11 divides 709 & not 13 divides 709 & not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )
709 = (3 * 236) + 1 ;
hence not 3 divides 709 by NAT_4:9; :: thesis: ( not 5 divides 709 & not 7 divides 709 & not 11 divides 709 & not 13 divides 709 & not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )
709 = (5 * 141) + 4 ;
hence not 5 divides 709 by NAT_4:9; :: thesis: ( not 7 divides 709 & not 11 divides 709 & not 13 divides 709 & not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )
709 = (7 * 101) + 2 ;
hence not 7 divides 709 by NAT_4:9; :: thesis: ( not 11 divides 709 & not 13 divides 709 & not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )
709 = (11 * 64) + 5 ;
hence not 11 divides 709 by NAT_4:9; :: thesis: ( not 13 divides 709 & not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )
709 = (13 * 54) + 7 ;
hence not 13 divides 709 by NAT_4:9; :: thesis: ( not 17 divides 709 & not 19 divides 709 & not 23 divides 709 )
709 = (17 * 41) + 12 ;
hence not 17 divides 709 by NAT_4:9; :: thesis: ( not 19 divides 709 & not 23 divides 709 )
709 = (19 * 37) + 6 ;
hence not 19 divides 709 by NAT_4:9; :: thesis: not 23 divides 709
709 = (23 * 30) + 19 ;
hence not 23 divides 709 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 709 & n is prime holds
not n divides 709 by XPRIMET1:18;
hence 709 is prime by NAT_4:14; :: thesis: verum