now :: thesis: ( not 2 divides 593 & not 3 divides 593 & not 5 divides 593 & not 7 divides 593 & not 11 divides 593 & not 13 divides 593 & not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )

593 = (2 * 296) + 1 ;
hence not 2 divides 593 by NAT_4:9; :: thesis: ( not 3 divides 593 & not 5 divides 593 & not 7 divides 593 & not 11 divides 593 & not 13 divides 593 & not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )
593 = (3 * 197) + 2 ;
hence not 3 divides 593 by NAT_4:9; :: thesis: ( not 5 divides 593 & not 7 divides 593 & not 11 divides 593 & not 13 divides 593 & not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )
593 = (5 * 118) + 3 ;
hence not 5 divides 593 by NAT_4:9; :: thesis: ( not 7 divides 593 & not 11 divides 593 & not 13 divides 593 & not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )
593 = (7 * 84) + 5 ;
hence not 7 divides 593 by NAT_4:9; :: thesis: ( not 11 divides 593 & not 13 divides 593 & not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )
593 = (11 * 53) + 10 ;
hence not 11 divides 593 by NAT_4:9; :: thesis: ( not 13 divides 593 & not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )
593 = (13 * 45) + 8 ;
hence not 13 divides 593 by NAT_4:9; :: thesis: ( not 17 divides 593 & not 19 divides 593 & not 23 divides 593 )
593 = (17 * 34) + 15 ;
hence not 17 divides 593 by NAT_4:9; :: thesis: ( not 19 divides 593 & not 23 divides 593 )
593 = (19 * 31) + 4 ;
hence not 19 divides 593 by NAT_4:9; :: thesis: not 23 divides 593
593 = (23 * 25) + 18 ;
hence not 23 divides 593 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 593 & n is prime holds
not n divides 593 by XPRIMET1:18;
hence 593 is prime by NAT_4:14; :: thesis: verum