now :: thesis: ( not 2 divides 751 & not 3 divides 751 & not 5 divides 751 & not 7 divides 751 & not 11 divides 751 & not 13 divides 751 & not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )

751 = (2 * 375) + 1 ;
hence not 2 divides 751 by NAT_4:9; :: thesis: ( not 3 divides 751 & not 5 divides 751 & not 7 divides 751 & not 11 divides 751 & not 13 divides 751 & not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )
751 = (3 * 250) + 1 ;
hence not 3 divides 751 by NAT_4:9; :: thesis: ( not 5 divides 751 & not 7 divides 751 & not 11 divides 751 & not 13 divides 751 & not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )
751 = (5 * 150) + 1 ;
hence not 5 divides 751 by NAT_4:9; :: thesis: ( not 7 divides 751 & not 11 divides 751 & not 13 divides 751 & not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )
751 = (7 * 107) + 2 ;
hence not 7 divides 751 by NAT_4:9; :: thesis: ( not 11 divides 751 & not 13 divides 751 & not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )
751 = (11 * 68) + 3 ;
hence not 11 divides 751 by NAT_4:9; :: thesis: ( not 13 divides 751 & not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )
751 = (13 * 57) + 10 ;
hence not 13 divides 751 by NAT_4:9; :: thesis: ( not 17 divides 751 & not 19 divides 751 & not 23 divides 751 )
751 = (17 * 44) + 3 ;
hence not 17 divides 751 by NAT_4:9; :: thesis: ( not 19 divides 751 & not 23 divides 751 )
751 = (19 * 39) + 10 ;
hence not 19 divides 751 by NAT_4:9; :: thesis: not 23 divides 751
751 = (23 * 32) + 15 ;
hence not 23 divides 751 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 751 & n is prime holds
not n divides 751 by XPRIMET1:18;
hence 751 is prime by NAT_4:14; :: thesis: verum