now :: thesis: ( not 2 divides 487 & not 3 divides 487 & not 5 divides 487 & not 7 divides 487 & not 11 divides 487 & not 13 divides 487 & not 17 divides 487 & not 19 divides 487 )

487 = (2 * 243) + 1 ;
hence not 2 divides 487 by NAT_4:9; :: thesis: ( not 3 divides 487 & not 5 divides 487 & not 7 divides 487 & not 11 divides 487 & not 13 divides 487 & not 17 divides 487 & not 19 divides 487 )
487 = (3 * 162) + 1 ;
hence not 3 divides 487 by NAT_4:9; :: thesis: ( not 5 divides 487 & not 7 divides 487 & not 11 divides 487 & not 13 divides 487 & not 17 divides 487 & not 19 divides 487 )
487 = (5 * 97) + 2 ;
hence not 5 divides 487 by NAT_4:9; :: thesis: ( not 7 divides 487 & not 11 divides 487 & not 13 divides 487 & not 17 divides 487 & not 19 divides 487 )
487 = (7 * 69) + 4 ;
hence not 7 divides 487 by NAT_4:9; :: thesis: ( not 11 divides 487 & not 13 divides 487 & not 17 divides 487 & not 19 divides 487 )
487 = (11 * 44) + 3 ;
hence not 11 divides 487 by NAT_4:9; :: thesis: ( not 13 divides 487 & not 17 divides 487 & not 19 divides 487 )
487 = (13 * 37) + 6 ;
hence not 13 divides 487 by NAT_4:9; :: thesis: ( not 17 divides 487 & not 19 divides 487 )
487 = (17 * 28) + 11 ;
hence not 17 divides 487 by NAT_4:9; :: thesis: not 19 divides 487
487 = (19 * 25) + 12 ;
hence not 19 divides 487 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 487 & n is prime holds
not n divides 487 by XPRIMET1:16;
hence 487 is prime by NAT_4:14; :: thesis: verum