now :: thesis: ( not 2 divides 1439 & not 3 divides 1439 & not 5 divides 1439 & not 7 divides 1439 & not 11 divides 1439 & not 13 divides 1439 & not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )

1439 = (2 * 719) + 1 ;
hence not 2 divides 1439 by NAT_4:9; :: thesis: ( not 3 divides 1439 & not 5 divides 1439 & not 7 divides 1439 & not 11 divides 1439 & not 13 divides 1439 & not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (3 * 479) + 2 ;
hence not 3 divides 1439 by NAT_4:9; :: thesis: ( not 5 divides 1439 & not 7 divides 1439 & not 11 divides 1439 & not 13 divides 1439 & not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (5 * 287) + 4 ;
hence not 5 divides 1439 by NAT_4:9; :: thesis: ( not 7 divides 1439 & not 11 divides 1439 & not 13 divides 1439 & not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (7 * 205) + 4 ;
hence not 7 divides 1439 by NAT_4:9; :: thesis: ( not 11 divides 1439 & not 13 divides 1439 & not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (11 * 130) + 9 ;
hence not 11 divides 1439 by NAT_4:9; :: thesis: ( not 13 divides 1439 & not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (13 * 110) + 9 ;
hence not 13 divides 1439 by NAT_4:9; :: thesis: ( not 17 divides 1439 & not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (17 * 84) + 11 ;
hence not 17 divides 1439 by NAT_4:9; :: thesis: ( not 19 divides 1439 & not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (19 * 75) + 14 ;
hence not 19 divides 1439 by NAT_4:9; :: thesis: ( not 23 divides 1439 & not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (23 * 62) + 13 ;
hence not 23 divides 1439 by NAT_4:9; :: thesis: ( not 29 divides 1439 & not 31 divides 1439 & not 37 divides 1439 )
1439 = (29 * 49) + 18 ;
hence not 29 divides 1439 by NAT_4:9; :: thesis: ( not 31 divides 1439 & not 37 divides 1439 )
1439 = (31 * 46) + 13 ;
hence not 31 divides 1439 by NAT_4:9; :: thesis: not 37 divides 1439
1439 = (37 * 38) + 33 ;
hence not 37 divides 1439 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1439 & n is prime holds
not n divides 1439 by XPRIMET1:24;
hence 1439 is prime by NAT_4:14; :: thesis: verum