now :: thesis: ( not 2 divides 1999 & not 3 divides 1999 & not 5 divides 1999 & not 7 divides 1999 & not 11 divides 1999 & not 13 divides 1999 & not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )

1999 = (2 * 999) + 1 ;
hence not 2 divides 1999 by NAT_4:9; :: thesis: ( not 3 divides 1999 & not 5 divides 1999 & not 7 divides 1999 & not 11 divides 1999 & not 13 divides 1999 & not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (3 * 666) + 1 ;
hence not 3 divides 1999 by NAT_4:9; :: thesis: ( not 5 divides 1999 & not 7 divides 1999 & not 11 divides 1999 & not 13 divides 1999 & not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (5 * 399) + 4 ;
hence not 5 divides 1999 by NAT_4:9; :: thesis: ( not 7 divides 1999 & not 11 divides 1999 & not 13 divides 1999 & not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (7 * 285) + 4 ;
hence not 7 divides 1999 by NAT_4:9; :: thesis: ( not 11 divides 1999 & not 13 divides 1999 & not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (11 * 181) + 8 ;
hence not 11 divides 1999 by NAT_4:9; :: thesis: ( not 13 divides 1999 & not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (13 * 153) + 10 ;
hence not 13 divides 1999 by NAT_4:9; :: thesis: ( not 17 divides 1999 & not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (17 * 117) + 10 ;
hence not 17 divides 1999 by NAT_4:9; :: thesis: ( not 19 divides 1999 & not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (19 * 105) + 4 ;
hence not 19 divides 1999 by NAT_4:9; :: thesis: ( not 23 divides 1999 & not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (23 * 86) + 21 ;
hence not 23 divides 1999 by NAT_4:9; :: thesis: ( not 29 divides 1999 & not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (29 * 68) + 27 ;
hence not 29 divides 1999 by NAT_4:9; :: thesis: ( not 31 divides 1999 & not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (31 * 64) + 15 ;
hence not 31 divides 1999 by NAT_4:9; :: thesis: ( not 37 divides 1999 & not 41 divides 1999 & not 43 divides 1999 )
1999 = (37 * 54) + 1 ;
hence not 37 divides 1999 by NAT_4:9; :: thesis: ( not 41 divides 1999 & not 43 divides 1999 )
1999 = (41 * 48) + 31 ;
hence not 41 divides 1999 by NAT_4:9; :: thesis: not 43 divides 1999
1999 = (43 * 46) + 21 ;
hence not 43 divides 1999 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1999 & n is prime holds
not n divides 1999 by XPRIMET1:28;
hence 1999 is prime by NAT_4:14; :: thesis: verum