now :: thesis: ( not 2 divides 733 & not 3 divides 733 & not 5 divides 733 & not 7 divides 733 & not 11 divides 733 & not 13 divides 733 & not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )

733 = (2 * 366) + 1 ;
hence not 2 divides 733 by NAT_4:9; :: thesis: ( not 3 divides 733 & not 5 divides 733 & not 7 divides 733 & not 11 divides 733 & not 13 divides 733 & not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )
733 = (3 * 244) + 1 ;
hence not 3 divides 733 by NAT_4:9; :: thesis: ( not 5 divides 733 & not 7 divides 733 & not 11 divides 733 & not 13 divides 733 & not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )
733 = (5 * 146) + 3 ;
hence not 5 divides 733 by NAT_4:9; :: thesis: ( not 7 divides 733 & not 11 divides 733 & not 13 divides 733 & not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )
733 = (7 * 104) + 5 ;
hence not 7 divides 733 by NAT_4:9; :: thesis: ( not 11 divides 733 & not 13 divides 733 & not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )
733 = (11 * 66) + 7 ;
hence not 11 divides 733 by NAT_4:9; :: thesis: ( not 13 divides 733 & not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )
733 = (13 * 56) + 5 ;
hence not 13 divides 733 by NAT_4:9; :: thesis: ( not 17 divides 733 & not 19 divides 733 & not 23 divides 733 )
733 = (17 * 43) + 2 ;
hence not 17 divides 733 by NAT_4:9; :: thesis: ( not 19 divides 733 & not 23 divides 733 )
733 = (19 * 38) + 11 ;
hence not 19 divides 733 by NAT_4:9; :: thesis: not 23 divides 733
733 = (23 * 31) + 20 ;
hence not 23 divides 733 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 733 & n is prime holds
not n divides 733 by XPRIMET1:18;
hence 733 is prime by NAT_4:14; :: thesis: verum