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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E12">Th12</font></span>: <a NAME="T12"><span class="comment"><font color="firebrick">:: LOPBAN_7:12</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b3">T</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> holds <br/> ( <font color="Olive" title="b3">T</font> is  <a href="lopban_7.html#V1" title="LOPBAN_7:attr.1">closed</a>  iff  for <font color="Olive" title="b4">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">T</font> &amp; <font color="Olive" title="b4">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b3">T</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b4">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  holds <br/>(  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Olive" title="b4">seq</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">T</font> &amp;  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Olive" title="b4">seq</font></span>)</span> ) )</div></div>
