now :: thesis: ( not 2 divides 541 & not 3 divides 541 & not 5 divides 541 & not 7 divides 541 & not 11 divides 541 & not 13 divides 541 & not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )

541 = (2 * 270) + 1 ;
hence not 2 divides 541 by NAT_4:9; :: thesis: ( not 3 divides 541 & not 5 divides 541 & not 7 divides 541 & not 11 divides 541 & not 13 divides 541 & not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )
541 = (3 * 180) + 1 ;
hence not 3 divides 541 by NAT_4:9; :: thesis: ( not 5 divides 541 & not 7 divides 541 & not 11 divides 541 & not 13 divides 541 & not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )
541 = (5 * 108) + 1 ;
hence not 5 divides 541 by NAT_4:9; :: thesis: ( not 7 divides 541 & not 11 divides 541 & not 13 divides 541 & not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )
541 = (7 * 77) + 2 ;
hence not 7 divides 541 by NAT_4:9; :: thesis: ( not 11 divides 541 & not 13 divides 541 & not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )
541 = (11 * 49) + 2 ;
hence not 11 divides 541 by NAT_4:9; :: thesis: ( not 13 divides 541 & not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )
541 = (13 * 41) + 8 ;
hence not 13 divides 541 by NAT_4:9; :: thesis: ( not 17 divides 541 & not 19 divides 541 & not 23 divides 541 )
541 = (17 * 31) + 14 ;
hence not 17 divides 541 by NAT_4:9; :: thesis: ( not 19 divides 541 & not 23 divides 541 )
541 = (19 * 28) + 9 ;
hence not 19 divides 541 by NAT_4:9; :: thesis: not 23 divides 541
541 = (23 * 23) + 12 ;
hence not 23 divides 541 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 541 & n is prime holds
not n divides 541 by XPRIMET1:18;
hence 541 is prime by NAT_4:14; :: thesis: verum