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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E32">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: TOPGEN_3:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  :  ex <font color="Olive" title="b3">U</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b3">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Olive" title="b3">U</font> ) </span> } </span>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> </div></div>
