now :: thesis: ( not 2 divides 587 & not 3 divides 587 & not 5 divides 587 & not 7 divides 587 & not 11 divides 587 & not 13 divides 587 & not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )

587 = (2 * 293) + 1 ;
hence not 2 divides 587 by NAT_4:9; :: thesis: ( not 3 divides 587 & not 5 divides 587 & not 7 divides 587 & not 11 divides 587 & not 13 divides 587 & not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )
587 = (3 * 195) + 2 ;
hence not 3 divides 587 by NAT_4:9; :: thesis: ( not 5 divides 587 & not 7 divides 587 & not 11 divides 587 & not 13 divides 587 & not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )
587 = (5 * 117) + 2 ;
hence not 5 divides 587 by NAT_4:9; :: thesis: ( not 7 divides 587 & not 11 divides 587 & not 13 divides 587 & not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )
587 = (7 * 83) + 6 ;
hence not 7 divides 587 by NAT_4:9; :: thesis: ( not 11 divides 587 & not 13 divides 587 & not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )
587 = (11 * 53) + 4 ;
hence not 11 divides 587 by NAT_4:9; :: thesis: ( not 13 divides 587 & not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )
587 = (13 * 45) + 2 ;
hence not 13 divides 587 by NAT_4:9; :: thesis: ( not 17 divides 587 & not 19 divides 587 & not 23 divides 587 )
587 = (17 * 34) + 9 ;
hence not 17 divides 587 by NAT_4:9; :: thesis: ( not 19 divides 587 & not 23 divides 587 )
587 = (19 * 30) + 17 ;
hence not 19 divides 587 by NAT_4:9; :: thesis: not 23 divides 587
587 = (23 * 25) + 12 ;
hence not 23 divides 587 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 587 & n is prime holds
not n divides 587 by XPRIMET1:18;
hence 587 is prime by NAT_4:14; :: thesis: verum