now :: thesis: ( not 2 divides 379 & not 3 divides 379 & not 5 divides 379 & not 7 divides 379 & not 11 divides 379 & not 13 divides 379 & not 17 divides 379 & not 19 divides 379 )

379 = (2 * 189) + 1 ;
hence not 2 divides 379 by NAT_4:9; :: thesis: ( not 3 divides 379 & not 5 divides 379 & not 7 divides 379 & not 11 divides 379 & not 13 divides 379 & not 17 divides 379 & not 19 divides 379 )
379 = (3 * 126) + 1 ;
hence not 3 divides 379 by NAT_4:9; :: thesis: ( not 5 divides 379 & not 7 divides 379 & not 11 divides 379 & not 13 divides 379 & not 17 divides 379 & not 19 divides 379 )
379 = (5 * 75) + 4 ;
hence not 5 divides 379 by NAT_4:9; :: thesis: ( not 7 divides 379 & not 11 divides 379 & not 13 divides 379 & not 17 divides 379 & not 19 divides 379 )
379 = (7 * 54) + 1 ;
hence not 7 divides 379 by NAT_4:9; :: thesis: ( not 11 divides 379 & not 13 divides 379 & not 17 divides 379 & not 19 divides 379 )
379 = (11 * 34) + 5 ;
hence not 11 divides 379 by NAT_4:9; :: thesis: ( not 13 divides 379 & not 17 divides 379 & not 19 divides 379 )
379 = (13 * 29) + 2 ;
hence not 13 divides 379 by NAT_4:9; :: thesis: ( not 17 divides 379 & not 19 divides 379 )
379 = (17 * 22) + 5 ;
hence not 17 divides 379 by NAT_4:9; :: thesis: not 19 divides 379
379 = (19 * 19) + 18 ;
hence not 19 divides 379 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 379 & n is prime holds
not n divides 379 by XPRIMET1:16;
hence 379 is prime by NAT_4:14; :: thesis: verum