now :: thesis: ( not 2 divides 1021 & not 3 divides 1021 & not 5 divides 1021 & not 7 divides 1021 & not 11 divides 1021 & not 13 divides 1021 & not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )

1021 = (2 * 510) + 1 ;
hence not 2 divides 1021 by NAT_4:9; :: thesis: ( not 3 divides 1021 & not 5 divides 1021 & not 7 divides 1021 & not 11 divides 1021 & not 13 divides 1021 & not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (3 * 340) + 1 ;
hence not 3 divides 1021 by NAT_4:9; :: thesis: ( not 5 divides 1021 & not 7 divides 1021 & not 11 divides 1021 & not 13 divides 1021 & not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (5 * 204) + 1 ;
hence not 5 divides 1021 by NAT_4:9; :: thesis: ( not 7 divides 1021 & not 11 divides 1021 & not 13 divides 1021 & not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (7 * 145) + 6 ;
hence not 7 divides 1021 by NAT_4:9; :: thesis: ( not 11 divides 1021 & not 13 divides 1021 & not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (11 * 92) + 9 ;
hence not 11 divides 1021 by NAT_4:9; :: thesis: ( not 13 divides 1021 & not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (13 * 78) + 7 ;
hence not 13 divides 1021 by NAT_4:9; :: thesis: ( not 17 divides 1021 & not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (17 * 60) + 1 ;
hence not 17 divides 1021 by NAT_4:9; :: thesis: ( not 19 divides 1021 & not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (19 * 53) + 14 ;
hence not 19 divides 1021 by NAT_4:9; :: thesis: ( not 23 divides 1021 & not 29 divides 1021 & not 31 divides 1021 )
1021 = (23 * 44) + 9 ;
hence not 23 divides 1021 by NAT_4:9; :: thesis: ( not 29 divides 1021 & not 31 divides 1021 )
1021 = (29 * 35) + 6 ;
hence not 29 divides 1021 by NAT_4:9; :: thesis: not 31 divides 1021
1021 = (31 * 32) + 29 ;
hence not 31 divides 1021 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1021 & n is prime holds
not n divides 1021 by XPRIMET1:22;
hence 1021 is prime by NAT_4:14; :: thesis: verum