now :: thesis: ( not 2 divides 37 & not 3 divides 37 & not 5 divides 37 )

37 = (2 * 18) + 1 ;
hence not 2 divides 37 by NAT_4:9; :: thesis: ( not 3 divides 37 & not 5 divides 37 )
37 = (3 * 12) + 1 ;
hence not 3 divides 37 by NAT_4:9; :: thesis: not 5 divides 37
37 = (5 * 7) + 2 ;
hence not 5 divides 37 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 37 & n is prime holds
not n divides 37 by XPRIMET1:6;
hence 37 is prime by NAT_4:14; :: thesis: verum