now :: thesis: ( not 2 divides 577 & not 3 divides 577 & not 5 divides 577 & not 7 divides 577 & not 11 divides 577 & not 13 divides 577 & not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )

577 = (2 * 288) + 1 ;
hence not 2 divides 577 by NAT_4:9; :: thesis: ( not 3 divides 577 & not 5 divides 577 & not 7 divides 577 & not 11 divides 577 & not 13 divides 577 & not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )
577 = (3 * 192) + 1 ;
hence not 3 divides 577 by NAT_4:9; :: thesis: ( not 5 divides 577 & not 7 divides 577 & not 11 divides 577 & not 13 divides 577 & not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )
577 = (5 * 115) + 2 ;
hence not 5 divides 577 by NAT_4:9; :: thesis: ( not 7 divides 577 & not 11 divides 577 & not 13 divides 577 & not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )
577 = (7 * 82) + 3 ;
hence not 7 divides 577 by NAT_4:9; :: thesis: ( not 11 divides 577 & not 13 divides 577 & not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )
577 = (11 * 52) + 5 ;
hence not 11 divides 577 by NAT_4:9; :: thesis: ( not 13 divides 577 & not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )
577 = (13 * 44) + 5 ;
hence not 13 divides 577 by NAT_4:9; :: thesis: ( not 17 divides 577 & not 19 divides 577 & not 23 divides 577 )
577 = (17 * 33) + 16 ;
hence not 17 divides 577 by NAT_4:9; :: thesis: ( not 19 divides 577 & not 23 divides 577 )
577 = (19 * 30) + 7 ;
hence not 19 divides 577 by NAT_4:9; :: thesis: not 23 divides 577
577 = (23 * 25) + 2 ;
hence not 23 divides 577 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 577 & n is prime holds
not n divides 577 by XPRIMET1:18;
hence 577 is prime by NAT_4:14; :: thesis: verum