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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E51">Th47</font></span>: <a NAME="T56"><span class="comment"><font color="firebrick">:: CARDFIL4:56</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">T</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><br/>  for <font color="Olive" title="b2">s</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">T</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">T</font> holds <br/> ( <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cardfil2.html#K18" title="CARDFIL2:func.18">lim_filter</a> (<font color="Olive" title="b2">s</font>,<span class="p1"><a href="cardfil4.html#K2" title="CARDFIL4:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>,<span class="p2">(<span class="default"><a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><a href="cardfil4.html#K2" title="CARDFIL4:func.2">.)</a></span>) iff  for <font color="Olive" title="b4">A</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b3">x</font>  ex <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st  <a href="cardfil4.html#K4" title="CARDFIL4:func.4">square-uparrow</a> <font color="Olive" title="b5">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b2">s</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Olive" title="b4">A</font> )</div></div>
