now :: thesis: ( not 2 divides 1051 & not 3 divides 1051 & not 5 divides 1051 & not 7 divides 1051 & not 11 divides 1051 & not 13 divides 1051 & not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )

1051 = (2 * 525) + 1 ;
hence not 2 divides 1051 by NAT_4:9; :: thesis: ( not 3 divides 1051 & not 5 divides 1051 & not 7 divides 1051 & not 11 divides 1051 & not 13 divides 1051 & not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (3 * 350) + 1 ;
hence not 3 divides 1051 by NAT_4:9; :: thesis: ( not 5 divides 1051 & not 7 divides 1051 & not 11 divides 1051 & not 13 divides 1051 & not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (5 * 210) + 1 ;
hence not 5 divides 1051 by NAT_4:9; :: thesis: ( not 7 divides 1051 & not 11 divides 1051 & not 13 divides 1051 & not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (7 * 150) + 1 ;
hence not 7 divides 1051 by NAT_4:9; :: thesis: ( not 11 divides 1051 & not 13 divides 1051 & not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (11 * 95) + 6 ;
hence not 11 divides 1051 by NAT_4:9; :: thesis: ( not 13 divides 1051 & not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (13 * 80) + 11 ;
hence not 13 divides 1051 by NAT_4:9; :: thesis: ( not 17 divides 1051 & not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (17 * 61) + 14 ;
hence not 17 divides 1051 by NAT_4:9; :: thesis: ( not 19 divides 1051 & not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (19 * 55) + 6 ;
hence not 19 divides 1051 by NAT_4:9; :: thesis: ( not 23 divides 1051 & not 29 divides 1051 & not 31 divides 1051 )
1051 = (23 * 45) + 16 ;
hence not 23 divides 1051 by NAT_4:9; :: thesis: ( not 29 divides 1051 & not 31 divides 1051 )
1051 = (29 * 36) + 7 ;
hence not 29 divides 1051 by NAT_4:9; :: thesis: not 31 divides 1051
1051 = (31 * 33) + 28 ;
hence not 31 divides 1051 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1051 & n is prime holds
not n divides 1051 by XPRIMET1:22;
hence 1051 is prime by NAT_4:14; :: thesis: verum