now :: thesis: ( not 2 divides 461 & not 3 divides 461 & not 5 divides 461 & not 7 divides 461 & not 11 divides 461 & not 13 divides 461 & not 17 divides 461 & not 19 divides 461 )

461 = (2 * 230) + 1 ;
hence not 2 divides 461 by NAT_4:9; :: thesis: ( not 3 divides 461 & not 5 divides 461 & not 7 divides 461 & not 11 divides 461 & not 13 divides 461 & not 17 divides 461 & not 19 divides 461 )
461 = (3 * 153) + 2 ;
hence not 3 divides 461 by NAT_4:9; :: thesis: ( not 5 divides 461 & not 7 divides 461 & not 11 divides 461 & not 13 divides 461 & not 17 divides 461 & not 19 divides 461 )
461 = (5 * 92) + 1 ;
hence not 5 divides 461 by NAT_4:9; :: thesis: ( not 7 divides 461 & not 11 divides 461 & not 13 divides 461 & not 17 divides 461 & not 19 divides 461 )
461 = (7 * 65) + 6 ;
hence not 7 divides 461 by NAT_4:9; :: thesis: ( not 11 divides 461 & not 13 divides 461 & not 17 divides 461 & not 19 divides 461 )
461 = (11 * 41) + 10 ;
hence not 11 divides 461 by NAT_4:9; :: thesis: ( not 13 divides 461 & not 17 divides 461 & not 19 divides 461 )
461 = (13 * 35) + 6 ;
hence not 13 divides 461 by NAT_4:9; :: thesis: ( not 17 divides 461 & not 19 divides 461 )
461 = (17 * 27) + 2 ;
hence not 17 divides 461 by NAT_4:9; :: thesis: not 19 divides 461
461 = (19 * 24) + 5 ;
hence not 19 divides 461 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 461 & n is prime holds
not n divides 461 by XPRIMET1:16;
hence 461 is prime by NAT_4:14; :: thesis: verum