now :: thesis: ( not 2 divides 821 & not 3 divides 821 & not 5 divides 821 & not 7 divides 821 & not 11 divides 821 & not 13 divides 821 & not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )

821 = (2 * 410) + 1 ;
hence not 2 divides 821 by NAT_4:9; :: thesis: ( not 3 divides 821 & not 5 divides 821 & not 7 divides 821 & not 11 divides 821 & not 13 divides 821 & not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )
821 = (3 * 273) + 2 ;
hence not 3 divides 821 by NAT_4:9; :: thesis: ( not 5 divides 821 & not 7 divides 821 & not 11 divides 821 & not 13 divides 821 & not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )
821 = (5 * 164) + 1 ;
hence not 5 divides 821 by NAT_4:9; :: thesis: ( not 7 divides 821 & not 11 divides 821 & not 13 divides 821 & not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )
821 = (7 * 117) + 2 ;
hence not 7 divides 821 by NAT_4:9; :: thesis: ( not 11 divides 821 & not 13 divides 821 & not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )
821 = (11 * 74) + 7 ;
hence not 11 divides 821 by NAT_4:9; :: thesis: ( not 13 divides 821 & not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )
821 = (13 * 63) + 2 ;
hence not 13 divides 821 by NAT_4:9; :: thesis: ( not 17 divides 821 & not 19 divides 821 & not 23 divides 821 )
821 = (17 * 48) + 5 ;
hence not 17 divides 821 by NAT_4:9; :: thesis: ( not 19 divides 821 & not 23 divides 821 )
821 = (19 * 43) + 4 ;
hence not 19 divides 821 by NAT_4:9; :: thesis: not 23 divides 821
821 = (23 * 35) + 16 ;
hence not 23 divides 821 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 821 & n is prime holds
not n divides 821 by XPRIMET1:18;
hence 821 is prime by NAT_4:14; :: thesis: verum