now :: thesis: ( not 2 divides 773 & not 3 divides 773 & not 5 divides 773 & not 7 divides 773 & not 11 divides 773 & not 13 divides 773 & not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )

773 = (2 * 386) + 1 ;
hence not 2 divides 773 by NAT_4:9; :: thesis: ( not 3 divides 773 & not 5 divides 773 & not 7 divides 773 & not 11 divides 773 & not 13 divides 773 & not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )
773 = (3 * 257) + 2 ;
hence not 3 divides 773 by NAT_4:9; :: thesis: ( not 5 divides 773 & not 7 divides 773 & not 11 divides 773 & not 13 divides 773 & not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )
773 = (5 * 154) + 3 ;
hence not 5 divides 773 by NAT_4:9; :: thesis: ( not 7 divides 773 & not 11 divides 773 & not 13 divides 773 & not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )
773 = (7 * 110) + 3 ;
hence not 7 divides 773 by NAT_4:9; :: thesis: ( not 11 divides 773 & not 13 divides 773 & not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )
773 = (11 * 70) + 3 ;
hence not 11 divides 773 by NAT_4:9; :: thesis: ( not 13 divides 773 & not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )
773 = (13 * 59) + 6 ;
hence not 13 divides 773 by NAT_4:9; :: thesis: ( not 17 divides 773 & not 19 divides 773 & not 23 divides 773 )
773 = (17 * 45) + 8 ;
hence not 17 divides 773 by NAT_4:9; :: thesis: ( not 19 divides 773 & not 23 divides 773 )
773 = (19 * 40) + 13 ;
hence not 19 divides 773 by NAT_4:9; :: thesis: not 23 divides 773
773 = (23 * 33) + 14 ;
hence not 23 divides 773 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 773 & n is prime holds
not n divides 773 by XPRIMET1:18;
hence 773 is prime by NAT_4:14; :: thesis: verum