now :: thesis: ( not 2 divides 827 & not 3 divides 827 & not 5 divides 827 & not 7 divides 827 & not 11 divides 827 & not 13 divides 827 & not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )

827 = (2 * 413) + 1 ;
hence not 2 divides 827 by NAT_4:9; :: thesis: ( not 3 divides 827 & not 5 divides 827 & not 7 divides 827 & not 11 divides 827 & not 13 divides 827 & not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )
827 = (3 * 275) + 2 ;
hence not 3 divides 827 by NAT_4:9; :: thesis: ( not 5 divides 827 & not 7 divides 827 & not 11 divides 827 & not 13 divides 827 & not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )
827 = (5 * 165) + 2 ;
hence not 5 divides 827 by NAT_4:9; :: thesis: ( not 7 divides 827 & not 11 divides 827 & not 13 divides 827 & not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )
827 = (7 * 118) + 1 ;
hence not 7 divides 827 by NAT_4:9; :: thesis: ( not 11 divides 827 & not 13 divides 827 & not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )
827 = (11 * 75) + 2 ;
hence not 11 divides 827 by NAT_4:9; :: thesis: ( not 13 divides 827 & not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )
827 = (13 * 63) + 8 ;
hence not 13 divides 827 by NAT_4:9; :: thesis: ( not 17 divides 827 & not 19 divides 827 & not 23 divides 827 )
827 = (17 * 48) + 11 ;
hence not 17 divides 827 by NAT_4:9; :: thesis: ( not 19 divides 827 & not 23 divides 827 )
827 = (19 * 43) + 10 ;
hence not 19 divides 827 by NAT_4:9; :: thesis: not 23 divides 827
827 = (23 * 35) + 22 ;
hence not 23 divides 827 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 827 & n is prime holds
not n divides 827 by XPRIMET1:18;
hence 827 is prime by NAT_4:14; :: thesis: verum