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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E31">Th30</font></span>: <a NAME="T23"><span class="comment"><font color="firebrick">:: RINGFRAC:23</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">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ringfrac.html#K6" title="RINGFRAC:func.6">Frac</a> <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">s</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b3">x</font>, <a href="ringfrac.html#K14" title="RINGFRAC:func.14">1.</a> (<font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">S</font>) <a href="ringfrac.html#R1" title="RINGFRAC:pred.1">Fr_Eq</a> <font color="Olive" title="b2">S</font></div></div>
