<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:35_1_1"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="metric_6.html#T10" target="_self" title="METRIC_6:th.10">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> )</span><br/>

<b>thus </b><a NAME="E4:35_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="txt" href="#E3:35_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="funct_2.html#T188" title="FUNCT_2:th.188">FUNCT_2:188</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> )</span><br/>

<b>hence </b><a NAME="E5:35_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> )</span><br/>

<b>thus </b><a NAME="E6:35_1_1"/>
<font color="Maroon" title="c3">S</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 <b>by </b><i><a class="txt" href="#E4:35_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="tbsp_1.html#T7" title="TBSP_1:th.7">TBSP_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">S</font> is <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a></span><br/>

<b>thus </b><a NAME="E7:35_1_1"/>
<font color="Maroon" title="c3">S</font> is <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a>
 <b>by </b><i><a class="txt" href="#E5:35_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="metric_6.html#T44" target="_self" title="METRIC_6:th.44">Th44</a>, <a class="ref" href="tbsp_1.html#T7" title="TBSP_1:th.7">TBSP_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
