811 = (2 * 405) + 1 ;
hence not 2 divides 811 by NAT_4:9; :: thesis:
811 = (3 * 270) + 1 ;
hence not 3 divides 811 by NAT_4:9; :: thesis:
811 = (5 * 162) + 1 ;
hence not 5 divides 811 by NAT_4:9; :: thesis:
811 = (7 * 115) + 6 ;
hence not 7 divides 811 by NAT_4:9; :: thesis:
811 = (11 * 73) + 8 ;
hence not 11 divides 811 by NAT_4:9; :: thesis:
811 = (13 * 62) + 5 ;
hence not 13 divides 811 by NAT_4:9; :: thesis:
811 = (17 * 47) + 12 ;
hence not 17 divides 811 by NAT_4:9; :: thesis:
811 = (19 * 42) + 13 ;
hence not 19 divides 811 by NAT_4:9; :: thesis:
811 = (23 * 35) + 6 ;
hence not 23 divides 811 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 811 & n is prime holds
not n divides 811 by XPRIMET1:18;
hence 811 is prime by NAT_4:14; :: thesis: