now :: thesis: ( not 2 divides 397 & not 3 divides 397 & not 5 divides 397 & not 7 divides 397 & not 11 divides 397 & not 13 divides 397 & not 17 divides 397 & not 19 divides 397 )

397 = (2 * 198) + 1 ;
hence not 2 divides 397 by NAT_4:9; :: thesis: ( not 3 divides 397 & not 5 divides 397 & not 7 divides 397 & not 11 divides 397 & not 13 divides 397 & not 17 divides 397 & not 19 divides 397 )
397 = (3 * 132) + 1 ;
hence not 3 divides 397 by NAT_4:9; :: thesis: ( not 5 divides 397 & not 7 divides 397 & not 11 divides 397 & not 13 divides 397 & not 17 divides 397 & not 19 divides 397 )
397 = (5 * 79) + 2 ;
hence not 5 divides 397 by NAT_4:9; :: thesis: ( not 7 divides 397 & not 11 divides 397 & not 13 divides 397 & not 17 divides 397 & not 19 divides 397 )
397 = (7 * 56) + 5 ;
hence not 7 divides 397 by NAT_4:9; :: thesis: ( not 11 divides 397 & not 13 divides 397 & not 17 divides 397 & not 19 divides 397 )
397 = (11 * 36) + 1 ;
hence not 11 divides 397 by NAT_4:9; :: thesis: ( not 13 divides 397 & not 17 divides 397 & not 19 divides 397 )
397 = (13 * 30) + 7 ;
hence not 13 divides 397 by NAT_4:9; :: thesis: ( not 17 divides 397 & not 19 divides 397 )
397 = (17 * 23) + 6 ;
hence not 17 divides 397 by NAT_4:9; :: thesis: not 19 divides 397
397 = (19 * 20) + 17 ;
hence not 19 divides 397 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 397 & n is prime holds
not n divides 397 by XPRIMET1:16;
hence 397 is prime by NAT_4:14; :: thesis: verum