now :: thesis: ( not 2 divides 1019 & not 3 divides 1019 & not 5 divides 1019 & not 7 divides 1019 & not 11 divides 1019 & not 13 divides 1019 & not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )

1019 = (2 * 509) + 1 ;
hence not 2 divides 1019 by NAT_4:9; :: thesis: ( not 3 divides 1019 & not 5 divides 1019 & not 7 divides 1019 & not 11 divides 1019 & not 13 divides 1019 & not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (3 * 339) + 2 ;
hence not 3 divides 1019 by NAT_4:9; :: thesis: ( not 5 divides 1019 & not 7 divides 1019 & not 11 divides 1019 & not 13 divides 1019 & not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (5 * 203) + 4 ;
hence not 5 divides 1019 by NAT_4:9; :: thesis: ( not 7 divides 1019 & not 11 divides 1019 & not 13 divides 1019 & not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (7 * 145) + 4 ;
hence not 7 divides 1019 by NAT_4:9; :: thesis: ( not 11 divides 1019 & not 13 divides 1019 & not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (11 * 92) + 7 ;
hence not 11 divides 1019 by NAT_4:9; :: thesis: ( not 13 divides 1019 & not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (13 * 78) + 5 ;
hence not 13 divides 1019 by NAT_4:9; :: thesis: ( not 17 divides 1019 & not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (17 * 59) + 16 ;
hence not 17 divides 1019 by NAT_4:9; :: thesis: ( not 19 divides 1019 & not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (19 * 53) + 12 ;
hence not 19 divides 1019 by NAT_4:9; :: thesis: ( not 23 divides 1019 & not 29 divides 1019 & not 31 divides 1019 )
1019 = (23 * 44) + 7 ;
hence not 23 divides 1019 by NAT_4:9; :: thesis: ( not 29 divides 1019 & not 31 divides 1019 )
1019 = (29 * 35) + 4 ;
hence not 29 divides 1019 by NAT_4:9; :: thesis: not 31 divides 1019
1019 = (31 * 32) + 27 ;
hence not 31 divides 1019 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1019 & n is prime holds
not n divides 1019 by XPRIMET1:22;
hence 1019 is prime by NAT_4:14; :: thesis: verum