now :: thesis: ( not 2 divides 701 & not 3 divides 701 & not 5 divides 701 & not 7 divides 701 & not 11 divides 701 & not 13 divides 701 & not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )

701 = (2 * 350) + 1 ;
hence not 2 divides 701 by NAT_4:9; :: thesis: ( not 3 divides 701 & not 5 divides 701 & not 7 divides 701 & not 11 divides 701 & not 13 divides 701 & not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )
701 = (3 * 233) + 2 ;
hence not 3 divides 701 by NAT_4:9; :: thesis: ( not 5 divides 701 & not 7 divides 701 & not 11 divides 701 & not 13 divides 701 & not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )
701 = (5 * 140) + 1 ;
hence not 5 divides 701 by NAT_4:9; :: thesis: ( not 7 divides 701 & not 11 divides 701 & not 13 divides 701 & not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )
701 = (7 * 100) + 1 ;
hence not 7 divides 701 by NAT_4:9; :: thesis: ( not 11 divides 701 & not 13 divides 701 & not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )
701 = (11 * 63) + 8 ;
hence not 11 divides 701 by NAT_4:9; :: thesis: ( not 13 divides 701 & not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )
701 = (13 * 53) + 12 ;
hence not 13 divides 701 by NAT_4:9; :: thesis: ( not 17 divides 701 & not 19 divides 701 & not 23 divides 701 )
701 = (17 * 41) + 4 ;
hence not 17 divides 701 by NAT_4:9; :: thesis: ( not 19 divides 701 & not 23 divides 701 )
701 = (19 * 36) + 17 ;
hence not 19 divides 701 by NAT_4:9; :: thesis: not 23 divides 701
701 = (23 * 30) + 11 ;
hence not 23 divides 701 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 701 & n is prime holds
not n divides 701 by XPRIMET1:18;
hence 701 is prime by NAT_4:14; :: thesis: verum