now :: thesis: ( not 2 divides 653 & not 3 divides 653 & not 5 divides 653 & not 7 divides 653 & not 11 divides 653 & not 13 divides 653 & not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )

653 = (2 * 326) + 1 ;
hence not 2 divides 653 by NAT_4:9; :: thesis: ( not 3 divides 653 & not 5 divides 653 & not 7 divides 653 & not 11 divides 653 & not 13 divides 653 & not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )
653 = (3 * 217) + 2 ;
hence not 3 divides 653 by NAT_4:9; :: thesis: ( not 5 divides 653 & not 7 divides 653 & not 11 divides 653 & not 13 divides 653 & not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )
653 = (5 * 130) + 3 ;
hence not 5 divides 653 by NAT_4:9; :: thesis: ( not 7 divides 653 & not 11 divides 653 & not 13 divides 653 & not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )
653 = (7 * 93) + 2 ;
hence not 7 divides 653 by NAT_4:9; :: thesis: ( not 11 divides 653 & not 13 divides 653 & not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )
653 = (11 * 59) + 4 ;
hence not 11 divides 653 by NAT_4:9; :: thesis: ( not 13 divides 653 & not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )
653 = (13 * 50) + 3 ;
hence not 13 divides 653 by NAT_4:9; :: thesis: ( not 17 divides 653 & not 19 divides 653 & not 23 divides 653 )
653 = (17 * 38) + 7 ;
hence not 17 divides 653 by NAT_4:9; :: thesis: ( not 19 divides 653 & not 23 divides 653 )
653 = (19 * 34) + 7 ;
hence not 19 divides 653 by NAT_4:9; :: thesis: not 23 divides 653
653 = (23 * 28) + 9 ;
hence not 23 divides 653 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 653 & n is prime holds
not n divides 653 by XPRIMET1:18;
hence 653 is prime by NAT_4:14; :: thesis: verum