now :: thesis: ( not 2 divides 607 & not 3 divides 607 & not 5 divides 607 & not 7 divides 607 & not 11 divides 607 & not 13 divides 607 & not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )

607 = (2 * 303) + 1 ;
hence not 2 divides 607 by NAT_4:9; :: thesis: ( not 3 divides 607 & not 5 divides 607 & not 7 divides 607 & not 11 divides 607 & not 13 divides 607 & not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )
607 = (3 * 202) + 1 ;
hence not 3 divides 607 by NAT_4:9; :: thesis: ( not 5 divides 607 & not 7 divides 607 & not 11 divides 607 & not 13 divides 607 & not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )
607 = (5 * 121) + 2 ;
hence not 5 divides 607 by NAT_4:9; :: thesis: ( not 7 divides 607 & not 11 divides 607 & not 13 divides 607 & not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )
607 = (7 * 86) + 5 ;
hence not 7 divides 607 by NAT_4:9; :: thesis: ( not 11 divides 607 & not 13 divides 607 & not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )
607 = (11 * 55) + 2 ;
hence not 11 divides 607 by NAT_4:9; :: thesis: ( not 13 divides 607 & not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )
607 = (13 * 46) + 9 ;
hence not 13 divides 607 by NAT_4:9; :: thesis: ( not 17 divides 607 & not 19 divides 607 & not 23 divides 607 )
607 = (17 * 35) + 12 ;
hence not 17 divides 607 by NAT_4:9; :: thesis: ( not 19 divides 607 & not 23 divides 607 )
607 = (19 * 31) + 18 ;
hence not 19 divides 607 by NAT_4:9; :: thesis: not 23 divides 607
607 = (23 * 26) + 9 ;
hence not 23 divides 607 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 607 & n is prime holds
not n divides 607 by XPRIMET1:18;
hence 607 is prime by NAT_4:14; :: thesis: verum