now :: thesis: ( not 2 divides 1061 & not 3 divides 1061 & not 5 divides 1061 & not 7 divides 1061 & not 11 divides 1061 & not 13 divides 1061 & not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )

1061 = (2 * 530) + 1 ;
hence not 2 divides 1061 by NAT_4:9; :: thesis: ( not 3 divides 1061 & not 5 divides 1061 & not 7 divides 1061 & not 11 divides 1061 & not 13 divides 1061 & not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (3 * 353) + 2 ;
hence not 3 divides 1061 by NAT_4:9; :: thesis: ( not 5 divides 1061 & not 7 divides 1061 & not 11 divides 1061 & not 13 divides 1061 & not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (5 * 212) + 1 ;
hence not 5 divides 1061 by NAT_4:9; :: thesis: ( not 7 divides 1061 & not 11 divides 1061 & not 13 divides 1061 & not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (7 * 151) + 4 ;
hence not 7 divides 1061 by NAT_4:9; :: thesis: ( not 11 divides 1061 & not 13 divides 1061 & not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (11 * 96) + 5 ;
hence not 11 divides 1061 by NAT_4:9; :: thesis: ( not 13 divides 1061 & not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (13 * 81) + 8 ;
hence not 13 divides 1061 by NAT_4:9; :: thesis: ( not 17 divides 1061 & not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (17 * 62) + 7 ;
hence not 17 divides 1061 by NAT_4:9; :: thesis: ( not 19 divides 1061 & not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (19 * 55) + 16 ;
hence not 19 divides 1061 by NAT_4:9; :: thesis: ( not 23 divides 1061 & not 29 divides 1061 & not 31 divides 1061 )
1061 = (23 * 46) + 3 ;
hence not 23 divides 1061 by NAT_4:9; :: thesis: ( not 29 divides 1061 & not 31 divides 1061 )
1061 = (29 * 36) + 17 ;
hence not 29 divides 1061 by NAT_4:9; :: thesis: not 31 divides 1061
1061 = (31 * 34) + 7 ;
hence not 31 divides 1061 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1061 & n is prime holds
not n divides 1061 by XPRIMET1:22;
hence 1061 is prime by NAT_4:14; :: thesis: verum