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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E30">Th47</font></span>: <a NAME="T47"><span class="comment"><font color="firebrick">:: QC_LANG3:47</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font> being    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-alphabet</a> <br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Olive" title="b1">A</font><br/>  for <font color="Olive" title="b3">V</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K2" title="QC_LANG1:func.2">QC-variables</a> <font color="Olive" title="b1">A</font></span>)</span>  st <font color="Olive" title="b2">p</font> is  <a href="qc_lang2.html#V2" title="QC_LANG2:attr.2">conditional</a>  holds <br/> <a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">V</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K12" title="QC_LANG2:func.12">the_antecedent_of</a> <font color="Olive" title="b2">p</font></span>)</span>,<font color="Olive" title="b3">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#NK13" title="QC_LANG2:NK.13">the_consequent_of</a> <font color="Olive" title="b2">p</font></span>)</span>,<font color="Olive" title="b3">V</font>)</span>)</span></div></div>
