now :: thesis: ( not 2 divides 113 & not 3 divides 113 & not 5 divides 113 & not 7 divides 113 )

113 = (2 * 56) + 1 ;
hence not 2 divides 113 by NAT_4:9; :: thesis: ( not 3 divides 113 & not 5 divides 113 & not 7 divides 113 )
113 = (3 * 37) + 2 ;
hence not 3 divides 113 by NAT_4:9; :: thesis: ( not 5 divides 113 & not 7 divides 113 )
113 = (5 * 22) + 3 ;
hence not 5 divides 113 by NAT_4:9; :: thesis: not 7 divides 113
113 = (7 * 16) + 1 ;
hence not 7 divides 113 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 113 & n is prime holds
not n divides 113 by XPRIMET1:8;
hence 113 is prime by NAT_4:14; :: thesis: verum