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<span class="kw">consider </span><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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:51_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#T10" target="_self" title="METRIC_6:th.10">Th10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E4:51_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c3">S</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="funct_2.html#T188" title="FUNCT_2:th.188">FUNCT_2:188</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">hence </span><a NAME="E5:51_1_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E6:51_1_1"/>
<font color="Maroon" title="c3">S</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="tbsp_1.html#T7" title="TBSP_1:th.7">TBSP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E7:51_1_1"/>
<font color="Maroon" title="c3">S</font> is  <a href="metric_6.html#V1" title="METRIC_6:attr.1">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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