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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E45">Th40</font></span>: <a NAME="T41"><span class="comment"><font color="firebrick">:: COMPL_SP:41</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">M</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>   <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>   <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> <br/>  for <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">M</font><br/>  for <font color="Olive" title="b4">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">WellSpace</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">X</font>)</span>)</span>  holds <br/>(  not <font color="Olive" title="b4">S</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>  or  for <font color="Olive" title="b5">Xa</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">WellSpace</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">X</font>)</span>)</span>  st <font color="Olive" title="b5">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> for <font color="Olive" title="b6">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b8">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b7">n</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b8">m</font></span>)</span>,<font color="Olive" title="b5">Xa</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">r</font> or  ex <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/> for <font color="Olive" title="b7">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b5">n</font> holds <br/> ex <font color="Olive" title="b8">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">M</font> st <font color="Olive" title="b4">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">Y</font>,<font color="Olive" title="b8">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</div></div>
