now :: thesis: ( not 2 divides 1013 & not 3 divides 1013 & not 5 divides 1013 & not 7 divides 1013 & not 11 divides 1013 & not 13 divides 1013 & not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )

1013 = (2 * 506) + 1 ;
hence not 2 divides 1013 by NAT_4:9; :: thesis: ( not 3 divides 1013 & not 5 divides 1013 & not 7 divides 1013 & not 11 divides 1013 & not 13 divides 1013 & not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (3 * 337) + 2 ;
hence not 3 divides 1013 by NAT_4:9; :: thesis: ( not 5 divides 1013 & not 7 divides 1013 & not 11 divides 1013 & not 13 divides 1013 & not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (5 * 202) + 3 ;
hence not 5 divides 1013 by NAT_4:9; :: thesis: ( not 7 divides 1013 & not 11 divides 1013 & not 13 divides 1013 & not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (7 * 144) + 5 ;
hence not 7 divides 1013 by NAT_4:9; :: thesis: ( not 11 divides 1013 & not 13 divides 1013 & not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (11 * 92) + 1 ;
hence not 11 divides 1013 by NAT_4:9; :: thesis: ( not 13 divides 1013 & not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (13 * 77) + 12 ;
hence not 13 divides 1013 by NAT_4:9; :: thesis: ( not 17 divides 1013 & not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (17 * 59) + 10 ;
hence not 17 divides 1013 by NAT_4:9; :: thesis: ( not 19 divides 1013 & not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (19 * 53) + 6 ;
hence not 19 divides 1013 by NAT_4:9; :: thesis: ( not 23 divides 1013 & not 29 divides 1013 & not 31 divides 1013 )
1013 = (23 * 44) + 1 ;
hence not 23 divides 1013 by NAT_4:9; :: thesis: ( not 29 divides 1013 & not 31 divides 1013 )
1013 = (29 * 34) + 27 ;
hence not 29 divides 1013 by NAT_4:9; :: thesis: not 31 divides 1013
1013 = (31 * 32) + 21 ;
hence not 31 divides 1013 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1013 & n is prime holds
not n divides 1013 by XPRIMET1:22;
hence 1013 is prime by NAT_4:14; :: thesis: verum