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<b>let </b><font color="Maroon">N</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="toprns_1.html#NM1">Real_Sequence</a> of <font color="Maroon">N</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">seq</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E45</font></i>: 
 <a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E3:57"/><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:57"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E5:57"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:57"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">n</font> = <font color="Maroon">n1</font>;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E8:57"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E9:57"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="toprns_1.html#E1:57"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E10:57"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:57"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:57"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:57"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:57"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
;<br/>
<a NAME="E14:57"/>
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
;<br/>
<b>hence </b><a NAME="E15:57"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#D1" target="_self">Def1</a>, <a class="ref" href="toprns_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


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