now :: thesis: ( not 2 divides 59 & not 3 divides 59 & not 5 divides 59 & not 7 divides 59 )

59 = (2 * 29) + 1 ;
hence not 2 divides 59 by NAT_4:9; :: thesis: ( not 3 divides 59 & not 5 divides 59 & not 7 divides 59 )
59 = (3 * 19) + 2 ;
hence not 3 divides 59 by NAT_4:9; :: thesis: ( not 5 divides 59 & not 7 divides 59 )
59 = (5 * 11) + 4 ;
hence not 5 divides 59 by NAT_4:9; :: thesis: not 7 divides 59
59 = (7 * 8) + 3 ;
hence not 7 divides 59 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 59 & n is prime holds
not n divides 59 by XPRIMET1:8;
hence 59 is prime by NAT_4:14; :: thesis: verum