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<div><span class="kw">theorem </span><a NAME="T16"><span class="comment"><font color="firebrick">:: CARDFIL3:16</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">N</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Olive" title="b1">N</font></span>)</span></span>)</span>  ex <font color="Olive" title="b3">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Olive" title="b1">N</font></span>)</span> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp;  <a href="frechet.html#K1" title="FRECHET:func.1">Balls</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">y</font>,<span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b4">n</font></span>)</span>)</span>)</span> where <font color="Olive" title="b4">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span>  ) <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#D1" onmouseover="rs('frechet/D1')" onmouseout="rh()">FRECHET:def 1</a></span>;<br/></div></div>
