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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:95"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0 as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E3:95"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:95"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> is   <a href="rat_1.html#NM1">Rational</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="prepower.html#NM1">Rational_Sequence</a> <b>by </b><i/>;<br/>
<a NAME="E6:95"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E7:95"/>
<font color="Maroon">s</font> <a href="prepower.html#K10" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E8:95"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><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/><a NAME="E1:95_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="prepower.html#T14" target="_self">Th14</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E10:95"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E11:95"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E12:95"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E13:95"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>;<br/>
<b>hence </b><a NAME="E14:95"/>
<font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> 0 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , , </i>;<br/>


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