now :: thesis: ( not 2 divides 617 & not 3 divides 617 & not 5 divides 617 & not 7 divides 617 & not 11 divides 617 & not 13 divides 617 & not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )

617 = (2 * 308) + 1 ;
hence not 2 divides 617 by NAT_4:9; :: thesis: ( not 3 divides 617 & not 5 divides 617 & not 7 divides 617 & not 11 divides 617 & not 13 divides 617 & not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )
617 = (3 * 205) + 2 ;
hence not 3 divides 617 by NAT_4:9; :: thesis: ( not 5 divides 617 & not 7 divides 617 & not 11 divides 617 & not 13 divides 617 & not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )
617 = (5 * 123) + 2 ;
hence not 5 divides 617 by NAT_4:9; :: thesis: ( not 7 divides 617 & not 11 divides 617 & not 13 divides 617 & not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )
617 = (7 * 88) + 1 ;
hence not 7 divides 617 by NAT_4:9; :: thesis: ( not 11 divides 617 & not 13 divides 617 & not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )
617 = (11 * 56) + 1 ;
hence not 11 divides 617 by NAT_4:9; :: thesis: ( not 13 divides 617 & not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )
617 = (13 * 47) + 6 ;
hence not 13 divides 617 by NAT_4:9; :: thesis: ( not 17 divides 617 & not 19 divides 617 & not 23 divides 617 )
617 = (17 * 36) + 5 ;
hence not 17 divides 617 by NAT_4:9; :: thesis: ( not 19 divides 617 & not 23 divides 617 )
617 = (19 * 32) + 9 ;
hence not 19 divides 617 by NAT_4:9; :: thesis: not 23 divides 617
617 = (23 * 26) + 19 ;
hence not 23 divides 617 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 617 & n is prime holds
not n divides 617 by XPRIMET1:18;
hence 617 is prime by NAT_4:14; :: thesis: verum