now :: thesis: ( not 2 divides 491 & not 3 divides 491 & not 5 divides 491 & not 7 divides 491 & not 11 divides 491 & not 13 divides 491 & not 17 divides 491 & not 19 divides 491 )

491 = (2 * 245) + 1 ;
hence not 2 divides 491 by NAT_4:9; :: thesis: ( not 3 divides 491 & not 5 divides 491 & not 7 divides 491 & not 11 divides 491 & not 13 divides 491 & not 17 divides 491 & not 19 divides 491 )
491 = (3 * 163) + 2 ;
hence not 3 divides 491 by NAT_4:9; :: thesis: ( not 5 divides 491 & not 7 divides 491 & not 11 divides 491 & not 13 divides 491 & not 17 divides 491 & not 19 divides 491 )
491 = (5 * 98) + 1 ;
hence not 5 divides 491 by NAT_4:9; :: thesis: ( not 7 divides 491 & not 11 divides 491 & not 13 divides 491 & not 17 divides 491 & not 19 divides 491 )
491 = (7 * 70) + 1 ;
hence not 7 divides 491 by NAT_4:9; :: thesis: ( not 11 divides 491 & not 13 divides 491 & not 17 divides 491 & not 19 divides 491 )
491 = (11 * 44) + 7 ;
hence not 11 divides 491 by NAT_4:9; :: thesis: ( not 13 divides 491 & not 17 divides 491 & not 19 divides 491 )
491 = (13 * 37) + 10 ;
hence not 13 divides 491 by NAT_4:9; :: thesis: ( not 17 divides 491 & not 19 divides 491 )
491 = (17 * 28) + 15 ;
hence not 17 divides 491 by NAT_4:9; :: thesis: not 19 divides 491
491 = (19 * 25) + 16 ;
hence not 19 divides 491 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 491 & n is prime holds
not n divides 491 by XPRIMET1:16;
hence 491 is prime by NAT_4:14; :: thesis: verum