now :: thesis: ( not 2 divides 53 & not 3 divides 53 & not 5 divides 53 & not 7 divides 53 )

53 = (2 * 26) + 1 ;
hence not 2 divides 53 by NAT_4:9; :: thesis: ( not 3 divides 53 & not 5 divides 53 & not 7 divides 53 )
53 = (3 * 17) + 2 ;
hence not 3 divides 53 by NAT_4:9; :: thesis: ( not 5 divides 53 & not 7 divides 53 )
53 = (5 * 10) + 3 ;
hence not 5 divides 53 by NAT_4:9; :: thesis: not 7 divides 53
53 = (7 * 7) + 4 ;
hence not 7 divides 53 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 53 & n is prime holds
not n divides 53 by XPRIMET1:8;
hence 53 is prime by NAT_4:14; :: thesis: verum