now :: thesis: ( not 2 divides 467 & not 3 divides 467 & not 5 divides 467 & not 7 divides 467 & not 11 divides 467 & not 13 divides 467 & not 17 divides 467 & not 19 divides 467 )

467 = (2 * 233) + 1 ;
hence not 2 divides 467 by NAT_4:9; :: thesis: ( not 3 divides 467 & not 5 divides 467 & not 7 divides 467 & not 11 divides 467 & not 13 divides 467 & not 17 divides 467 & not 19 divides 467 )
467 = (3 * 155) + 2 ;
hence not 3 divides 467 by NAT_4:9; :: thesis: ( not 5 divides 467 & not 7 divides 467 & not 11 divides 467 & not 13 divides 467 & not 17 divides 467 & not 19 divides 467 )
467 = (5 * 93) + 2 ;
hence not 5 divides 467 by NAT_4:9; :: thesis: ( not 7 divides 467 & not 11 divides 467 & not 13 divides 467 & not 17 divides 467 & not 19 divides 467 )
467 = (7 * 66) + 5 ;
hence not 7 divides 467 by NAT_4:9; :: thesis: ( not 11 divides 467 & not 13 divides 467 & not 17 divides 467 & not 19 divides 467 )
467 = (11 * 42) + 5 ;
hence not 11 divides 467 by NAT_4:9; :: thesis: ( not 13 divides 467 & not 17 divides 467 & not 19 divides 467 )
467 = (13 * 35) + 12 ;
hence not 13 divides 467 by NAT_4:9; :: thesis: ( not 17 divides 467 & not 19 divides 467 )
467 = (17 * 27) + 8 ;
hence not 17 divides 467 by NAT_4:9; :: thesis: not 19 divides 467
467 = (19 * 24) + 11 ;
hence not 19 divides 467 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 467 & n is prime holds
not n divides 467 by XPRIMET1:16;
hence 467 is prime by NAT_4:14; :: thesis: verum