now :: thesis: ( not 2 divides 929 & not 3 divides 929 & not 5 divides 929 & not 7 divides 929 & not 11 divides 929 & not 13 divides 929 & not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )

929 = (2 * 464) + 1 ;
hence not 2 divides 929 by NAT_4:9; :: thesis: ( not 3 divides 929 & not 5 divides 929 & not 7 divides 929 & not 11 divides 929 & not 13 divides 929 & not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (3 * 309) + 2 ;
hence not 3 divides 929 by NAT_4:9; :: thesis: ( not 5 divides 929 & not 7 divides 929 & not 11 divides 929 & not 13 divides 929 & not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (5 * 185) + 4 ;
hence not 5 divides 929 by NAT_4:9; :: thesis: ( not 7 divides 929 & not 11 divides 929 & not 13 divides 929 & not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (7 * 132) + 5 ;
hence not 7 divides 929 by NAT_4:9; :: thesis: ( not 11 divides 929 & not 13 divides 929 & not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (11 * 84) + 5 ;
hence not 11 divides 929 by NAT_4:9; :: thesis: ( not 13 divides 929 & not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (13 * 71) + 6 ;
hence not 13 divides 929 by NAT_4:9; :: thesis: ( not 17 divides 929 & not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (17 * 54) + 11 ;
hence not 17 divides 929 by NAT_4:9; :: thesis: ( not 19 divides 929 & not 23 divides 929 & not 29 divides 929 )
929 = (19 * 48) + 17 ;
hence not 19 divides 929 by NAT_4:9; :: thesis: ( not 23 divides 929 & not 29 divides 929 )
929 = (23 * 40) + 9 ;
hence not 23 divides 929 by NAT_4:9; :: thesis: not 29 divides 929
929 = (29 * 32) + 1 ;
hence not 29 divides 929 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 929 & n is prime holds
not n divides 929 by XPRIMET1:20;
hence 929 is prime by NAT_4:14; :: thesis: verum