now :: thesis: ( not 2 divides 389 & not 3 divides 389 & not 5 divides 389 & not 7 divides 389 & not 11 divides 389 & not 13 divides 389 & not 17 divides 389 & not 19 divides 389 )

389 = (2 * 194) + 1 ;
hence not 2 divides 389 by NAT_4:9; :: thesis: ( not 3 divides 389 & not 5 divides 389 & not 7 divides 389 & not 11 divides 389 & not 13 divides 389 & not 17 divides 389 & not 19 divides 389 )
389 = (3 * 129) + 2 ;
hence not 3 divides 389 by NAT_4:9; :: thesis: ( not 5 divides 389 & not 7 divides 389 & not 11 divides 389 & not 13 divides 389 & not 17 divides 389 & not 19 divides 389 )
389 = (5 * 77) + 4 ;
hence not 5 divides 389 by NAT_4:9; :: thesis: ( not 7 divides 389 & not 11 divides 389 & not 13 divides 389 & not 17 divides 389 & not 19 divides 389 )
389 = (7 * 55) + 4 ;
hence not 7 divides 389 by NAT_4:9; :: thesis: ( not 11 divides 389 & not 13 divides 389 & not 17 divides 389 & not 19 divides 389 )
389 = (11 * 35) + 4 ;
hence not 11 divides 389 by NAT_4:9; :: thesis: ( not 13 divides 389 & not 17 divides 389 & not 19 divides 389 )
389 = (13 * 29) + 12 ;
hence not 13 divides 389 by NAT_4:9; :: thesis: ( not 17 divides 389 & not 19 divides 389 )
389 = (17 * 22) + 15 ;
hence not 17 divides 389 by NAT_4:9; :: thesis: not 19 divides 389
389 = (19 * 20) + 9 ;
hence not 19 divides 389 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 389 & n is prime holds
not n divides 389 by XPRIMET1:16;
hence 389 is prime by NAT_4:14; :: thesis: verum