now :: thesis: ( not 2 divides 359 & not 3 divides 359 & not 5 divides 359 & not 7 divides 359 & not 11 divides 359 & not 13 divides 359 & not 17 divides 359 )

359 = (2 * 179) + 1 ;
hence not 2 divides 359 by NAT_4:9; :: thesis: ( not 3 divides 359 & not 5 divides 359 & not 7 divides 359 & not 11 divides 359 & not 13 divides 359 & not 17 divides 359 )
359 = (3 * 119) + 2 ;
hence not 3 divides 359 by NAT_4:9; :: thesis: ( not 5 divides 359 & not 7 divides 359 & not 11 divides 359 & not 13 divides 359 & not 17 divides 359 )
359 = (5 * 71) + 4 ;
hence not 5 divides 359 by NAT_4:9; :: thesis: ( not 7 divides 359 & not 11 divides 359 & not 13 divides 359 & not 17 divides 359 )
359 = (7 * 51) + 2 ;
hence not 7 divides 359 by NAT_4:9; :: thesis: ( not 11 divides 359 & not 13 divides 359 & not 17 divides 359 )
359 = (11 * 32) + 7 ;
hence not 11 divides 359 by NAT_4:9; :: thesis: ( not 13 divides 359 & not 17 divides 359 )
359 = (13 * 27) + 8 ;
hence not 13 divides 359 by NAT_4:9; :: thesis: not 17 divides 359
359 = (17 * 21) + 2 ;
hence not 17 divides 359 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 359 & n is prime holds
not n divides 359 by XPRIMET1:14;
hence 359 is prime by NAT_4:14; :: thesis: verum