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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E31">Th32</font></span>: <a NAME="T32"><span class="comment"><font color="firebrick">:: TOPDIM_1:32</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">T</font> being   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>  st <font color="Olive" title="b1">T</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>  &amp; (  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">T</font>  st <font color="Olive" title="b2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">B</font> holds <br/> ex <font color="Olive" title="b4">A9</font>, <font color="Olive" title="b5">B9</font> being   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">T</font> st <br/>( <font color="Olive" title="b4">A9</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b5">B9</font> &amp; <font color="Olive" title="b4">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b5">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">T</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">A9</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">B9</font> ) ) holds <br/>( <font color="Olive" title="b1">T</font> is  <a href="topdim_1.html#NV6" title="TOPDIM_1:NV.6">finite-ind</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Olive" title="b1">T</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</div></div>
