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<div><span class="kw">theorem </span><a NAME="T5"><span class="comment"><font color="firebrick">:: MODCAT_1:5</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b2">UN</font> being   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  ex <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">G</font>,<font color="Olive" title="b5">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b4">G</font> is    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">GroupObjects</a> <font color="Olive" title="b2">UN</font>, <font color="Olive" title="b5">f</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) : verum </span> } </span>  &amp;  <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Olive" title="b3">x</font>, <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b1">R</font> )</div></div>
