now :: thesis: ( not 2 divides 599 & not 3 divides 599 & not 5 divides 599 & not 7 divides 599 & not 11 divides 599 & not 13 divides 599 & not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )

599 = (2 * 299) + 1 ;
hence not 2 divides 599 by NAT_4:9; :: thesis: ( not 3 divides 599 & not 5 divides 599 & not 7 divides 599 & not 11 divides 599 & not 13 divides 599 & not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )
599 = (3 * 199) + 2 ;
hence not 3 divides 599 by NAT_4:9; :: thesis: ( not 5 divides 599 & not 7 divides 599 & not 11 divides 599 & not 13 divides 599 & not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )
599 = (5 * 119) + 4 ;
hence not 5 divides 599 by NAT_4:9; :: thesis: ( not 7 divides 599 & not 11 divides 599 & not 13 divides 599 & not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )
599 = (7 * 85) + 4 ;
hence not 7 divides 599 by NAT_4:9; :: thesis: ( not 11 divides 599 & not 13 divides 599 & not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )
599 = (11 * 54) + 5 ;
hence not 11 divides 599 by NAT_4:9; :: thesis: ( not 13 divides 599 & not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )
599 = (13 * 46) + 1 ;
hence not 13 divides 599 by NAT_4:9; :: thesis: ( not 17 divides 599 & not 19 divides 599 & not 23 divides 599 )
599 = (17 * 35) + 4 ;
hence not 17 divides 599 by NAT_4:9; :: thesis: ( not 19 divides 599 & not 23 divides 599 )
599 = (19 * 31) + 10 ;
hence not 19 divides 599 by NAT_4:9; :: thesis: not 23 divides 599
599 = (23 * 26) + 1 ;
hence not 23 divides 599 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 599 & n is prime holds
not n divides 599 by XPRIMET1:18;
hence 599 is prime by NAT_4:14; :: thesis: verum