now :: thesis: ( not 2 divides 613 & not 3 divides 613 & not 5 divides 613 & not 7 divides 613 & not 11 divides 613 & not 13 divides 613 & not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )

613 = (2 * 306) + 1 ;
hence not 2 divides 613 by NAT_4:9; :: thesis: ( not 3 divides 613 & not 5 divides 613 & not 7 divides 613 & not 11 divides 613 & not 13 divides 613 & not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )
613 = (3 * 204) + 1 ;
hence not 3 divides 613 by NAT_4:9; :: thesis: ( not 5 divides 613 & not 7 divides 613 & not 11 divides 613 & not 13 divides 613 & not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )
613 = (5 * 122) + 3 ;
hence not 5 divides 613 by NAT_4:9; :: thesis: ( not 7 divides 613 & not 11 divides 613 & not 13 divides 613 & not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )
613 = (7 * 87) + 4 ;
hence not 7 divides 613 by NAT_4:9; :: thesis: ( not 11 divides 613 & not 13 divides 613 & not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )
613 = (11 * 55) + 8 ;
hence not 11 divides 613 by NAT_4:9; :: thesis: ( not 13 divides 613 & not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )
613 = (13 * 47) + 2 ;
hence not 13 divides 613 by NAT_4:9; :: thesis: ( not 17 divides 613 & not 19 divides 613 & not 23 divides 613 )
613 = (17 * 36) + 1 ;
hence not 17 divides 613 by NAT_4:9; :: thesis: ( not 19 divides 613 & not 23 divides 613 )
613 = (19 * 32) + 5 ;
hence not 19 divides 613 by NAT_4:9; :: thesis: not 23 divides 613
613 = (23 * 26) + 15 ;
hence not 23 divides 613 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 613 & n is prime holds
not n divides 613 by XPRIMET1:18;
hence 613 is prime by NAT_4:14; :: thesis: verum