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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E55">exfin</font></span>: <a NAME="T64"><span class="comment"><font color="firebrick">:: PL_AXIOM:64</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">F</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="pl_axiom.html#K1" title="PL_AXIOM:func.1">PL-WFF</a><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="pl_axiom.html#K1" title="PL_AXIOM:func.1">PL-WFF</a>  holds <br/> ( <font color="Olive" title="b1">F</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <font color="Olive" title="b2">A</font> iff  ex <font color="Olive" title="b3">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="pl_axiom.html#K1" title="PL_AXIOM:func.1">PL-WFF</a> st <br/>( <font color="Olive" title="b3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b3">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Olive" title="b3">G</font> <a href="pl_axiom.html#R6" title="PL_AXIOM:pred.6">|-</a> <font color="Olive" title="b2">A</font> ) )</div></div>
