<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E6">Th6</font></span>: <a NAME="T6"><span class="comment"><font color="firebrick">:: TBSP_1:6</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">N</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> <br/>  for <font color="Olive" title="b2">S2</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">N</font> holds <br/> ( <font color="Olive" title="b2">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>  iff  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">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="b4">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">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="b2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> )</div></div>
