now :: thesis: ( not 2 divides 739 & not 3 divides 739 & not 5 divides 739 & not 7 divides 739 & not 11 divides 739 & not 13 divides 739 & not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )

739 = (2 * 369) + 1 ;
hence not 2 divides 739 by NAT_4:9; :: thesis: ( not 3 divides 739 & not 5 divides 739 & not 7 divides 739 & not 11 divides 739 & not 13 divides 739 & not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )
739 = (3 * 246) + 1 ;
hence not 3 divides 739 by NAT_4:9; :: thesis: ( not 5 divides 739 & not 7 divides 739 & not 11 divides 739 & not 13 divides 739 & not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )
739 = (5 * 147) + 4 ;
hence not 5 divides 739 by NAT_4:9; :: thesis: ( not 7 divides 739 & not 11 divides 739 & not 13 divides 739 & not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )
739 = (7 * 105) + 4 ;
hence not 7 divides 739 by NAT_4:9; :: thesis: ( not 11 divides 739 & not 13 divides 739 & not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )
739 = (11 * 67) + 2 ;
hence not 11 divides 739 by NAT_4:9; :: thesis: ( not 13 divides 739 & not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )
739 = (13 * 56) + 11 ;
hence not 13 divides 739 by NAT_4:9; :: thesis: ( not 17 divides 739 & not 19 divides 739 & not 23 divides 739 )
739 = (17 * 43) + 8 ;
hence not 17 divides 739 by NAT_4:9; :: thesis: ( not 19 divides 739 & not 23 divides 739 )
739 = (19 * 38) + 17 ;
hence not 19 divides 739 by NAT_4:9; :: thesis: not 23 divides 739
739 = (23 * 32) + 3 ;
hence not 23 divides 739 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 739 & n is prime holds
not n divides 739 by XPRIMET1:18;
hence 739 is prime by NAT_4:14; :: thesis: verum