now :: thesis: ( not 2 divides 1451 & not 3 divides 1451 & not 5 divides 1451 & not 7 divides 1451 & not 11 divides 1451 & not 13 divides 1451 & not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )

1451 = (2 * 725) + 1 ;
hence not 2 divides 1451 by NAT_4:9; :: thesis: ( not 3 divides 1451 & not 5 divides 1451 & not 7 divides 1451 & not 11 divides 1451 & not 13 divides 1451 & not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (3 * 483) + 2 ;
hence not 3 divides 1451 by NAT_4:9; :: thesis: ( not 5 divides 1451 & not 7 divides 1451 & not 11 divides 1451 & not 13 divides 1451 & not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (5 * 290) + 1 ;
hence not 5 divides 1451 by NAT_4:9; :: thesis: ( not 7 divides 1451 & not 11 divides 1451 & not 13 divides 1451 & not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (7 * 207) + 2 ;
hence not 7 divides 1451 by NAT_4:9; :: thesis: ( not 11 divides 1451 & not 13 divides 1451 & not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (11 * 131) + 10 ;
hence not 11 divides 1451 by NAT_4:9; :: thesis: ( not 13 divides 1451 & not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (13 * 111) + 8 ;
hence not 13 divides 1451 by NAT_4:9; :: thesis: ( not 17 divides 1451 & not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (17 * 85) + 6 ;
hence not 17 divides 1451 by NAT_4:9; :: thesis: ( not 19 divides 1451 & not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (19 * 76) + 7 ;
hence not 19 divides 1451 by NAT_4:9; :: thesis: ( not 23 divides 1451 & not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (23 * 63) + 2 ;
hence not 23 divides 1451 by NAT_4:9; :: thesis: ( not 29 divides 1451 & not 31 divides 1451 & not 37 divides 1451 )
1451 = (29 * 50) + 1 ;
hence not 29 divides 1451 by NAT_4:9; :: thesis: ( not 31 divides 1451 & not 37 divides 1451 )
1451 = (31 * 46) + 25 ;
hence not 31 divides 1451 by NAT_4:9; :: thesis: not 37 divides 1451
1451 = (37 * 39) + 8 ;
hence not 37 divides 1451 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1451 & n is prime holds
not n divides 1451 by XPRIMET1:24;
hence 1451 is prime by NAT_4:14; :: thesis: verum