now :: thesis: ( not 2 divides 727 & not 3 divides 727 & not 5 divides 727 & not 7 divides 727 & not 11 divides 727 & not 13 divides 727 & not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )

727 = (2 * 363) + 1 ;
hence not 2 divides 727 by NAT_4:9; :: thesis: ( not 3 divides 727 & not 5 divides 727 & not 7 divides 727 & not 11 divides 727 & not 13 divides 727 & not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )
727 = (3 * 242) + 1 ;
hence not 3 divides 727 by NAT_4:9; :: thesis: ( not 5 divides 727 & not 7 divides 727 & not 11 divides 727 & not 13 divides 727 & not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )
727 = (5 * 145) + 2 ;
hence not 5 divides 727 by NAT_4:9; :: thesis: ( not 7 divides 727 & not 11 divides 727 & not 13 divides 727 & not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )
727 = (7 * 103) + 6 ;
hence not 7 divides 727 by NAT_4:9; :: thesis: ( not 11 divides 727 & not 13 divides 727 & not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )
727 = (11 * 66) + 1 ;
hence not 11 divides 727 by NAT_4:9; :: thesis: ( not 13 divides 727 & not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )
727 = (13 * 55) + 12 ;
hence not 13 divides 727 by NAT_4:9; :: thesis: ( not 17 divides 727 & not 19 divides 727 & not 23 divides 727 )
727 = (17 * 42) + 13 ;
hence not 17 divides 727 by NAT_4:9; :: thesis: ( not 19 divides 727 & not 23 divides 727 )
727 = (19 * 38) + 5 ;
hence not 19 divides 727 by NAT_4:9; :: thesis: not 23 divides 727
727 = (23 * 31) + 14 ;
hence not 23 divides 727 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 727 & n is prime holds
not n divides 727 by XPRIMET1:18;
hence 727 is prime by NAT_4:14; :: thesis: verum