  Equivalence in physics is discussed on the basis of experimental data
accompanied by experimental errors. The introduction of the equivalence being
consistent with the mathematical definition is possible only in theories
constructed on non-standard number spaces by taking the experimental errors as
infinitesimal numbers of the non-standard spaces. Following the idea for the
equivalence (the physical equivalence), a new description of space-time in
terms of infinitesimal-lattice points on non-standard real number space $\SR$
is proposed. The infinitesimal-lattice space, $^*{\cal L}$, is represented by
the set of points on $\SR$ which are written by $l_n=n\SE$, where the
infinitesimal lattice-spacing $\SE$ is determined by a non-standard natural
number $^*N$ such that $\SE\equiv ^*N^{-1}$. By using infinitesimal
neighborhoos ($\MON$) of real number $r$ on $\SL$ we can make a space $\SM$
which is isomorphic to $\RE$ as additive group. Therefore, every point on
$(\SM)^N$ automatically has the internal confined-subspace $\MON$. A field
theory on $\SL$ is proposed. To determine a projection from $\SL$ to $\SM$, a
fundamental principle based on the physical equivalence is introduced. The
physical equivalence is expressed by the totally equal treatment for
indistinguishable quantities in our observations. Following the principle, we
show that U(1) and SU(N) symmetries on the space $(\SM)^N$ are induced from the
internal substructure $(\MON)^N$. Quantized state describing configuration
space is constructed on $(\SM)^N$. We see that Lorentz and general relativistic
transformations are also represented by operators which involve the U(1) and
SU(N) internal symmetries.
