<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E49">Th46</font></span>: <a NAME="T30"><span class="comment"><font color="firebrick">:: RINGFRAC:30</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">R</font><br/>  for <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">S</font> <a href="ringfrac.html#NK17" title="RINGFRAC:NK.17">~</a> <font color="Olive" title="b1">R</font></span>)</span>  ex <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b5">r2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="ringfrac.html#K12" title="RINGFRAC:func.12">EqRel</a> <font color="Olive" title="b2">S</font></span>)</span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">r2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) )</div></div>
