now :: thesis: ( not 2 divides 2239 & not 3 divides 2239 & not 5 divides 2239 & not 7 divides 2239 & not 11 divides 2239 & not 13 divides 2239 & not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )

2239 = (2 * 1119) + 1 ;
hence not 2 divides 2239 by NAT_4:9; :: thesis: ( not 3 divides 2239 & not 5 divides 2239 & not 7 divides 2239 & not 11 divides 2239 & not 13 divides 2239 & not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (3 * 746) + 1 ;
hence not 3 divides 2239 by NAT_4:9; :: thesis: ( not 5 divides 2239 & not 7 divides 2239 & not 11 divides 2239 & not 13 divides 2239 & not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (5 * 447) + 4 ;
hence not 5 divides 2239 by NAT_4:9; :: thesis: ( not 7 divides 2239 & not 11 divides 2239 & not 13 divides 2239 & not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (7 * 319) + 6 ;
hence not 7 divides 2239 by NAT_4:9; :: thesis: ( not 11 divides 2239 & not 13 divides 2239 & not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (11 * 203) + 6 ;
hence not 11 divides 2239 by NAT_4:9; :: thesis: ( not 13 divides 2239 & not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (13 * 172) + 3 ;
hence not 13 divides 2239 by NAT_4:9; :: thesis: ( not 17 divides 2239 & not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (17 * 131) + 12 ;
hence not 17 divides 2239 by NAT_4:9; :: thesis: ( not 19 divides 2239 & not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (19 * 117) + 16 ;
hence not 19 divides 2239 by NAT_4:9; :: thesis: ( not 23 divides 2239 & not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (23 * 97) + 8 ;
hence not 23 divides 2239 by NAT_4:9; :: thesis: ( not 29 divides 2239 & not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (29 * 77) + 6 ;
hence not 29 divides 2239 by NAT_4:9; :: thesis: ( not 31 divides 2239 & not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (31 * 72) + 7 ;
hence not 31 divides 2239 by NAT_4:9; :: thesis: ( not 37 divides 2239 & not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (37 * 60) + 19 ;
hence not 37 divides 2239 by NAT_4:9; :: thesis: ( not 41 divides 2239 & not 43 divides 2239 & not 47 divides 2239 )
2239 = (41 * 54) + 25 ;
hence not 41 divides 2239 by NAT_4:9; :: thesis: ( not 43 divides 2239 & not 47 divides 2239 )
2239 = (43 * 52) + 3 ;
hence not 43 divides 2239 by NAT_4:9; :: thesis: not 47 divides 2239
2239 = (47 * 47) + 30 ;
hence not 47 divides 2239 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2239 & n is prime holds
not n divides 2239 by XPRIMET1:30;
hence 2239 is prime by NAT_4:14; :: thesis: verum