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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>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</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="E1:99"/><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>consider </b><font color="Maroon">sb</font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:99"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">sb</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">sb</font> &amp; ( 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">sb</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:99"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sb</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">sc</font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E6:99"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">sc</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">sc</font> &amp; ( 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">sc</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:99"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sc</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_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="E1:99_1"/>
<span class="p1">(<span class="default"><font color="Maroon">sb</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">sc</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sb</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">sc</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/><b>hence </b><a NAME="E2:99_1"/>
<span class="p1">(<span class="default"><font color="Maroon">sb</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">sc</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is   <a href="rat_1.html#NM1">Rational</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">sb</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">sc</font> as   <a href="prepower.html#NM1">Rational_Sequence</a> <b>by </b><i/>;<br/>
<a NAME="E10:99"/><i><font color="Green">E43</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_2.html#T19">SEQ_2:19</a></i>;<br/>
<a NAME="E11:99"/><i><font color="Green">E50</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>;<br/>
<a NAME="E12:99"/><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>, , </i>;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><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/><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/>.= 
<font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sb</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">sc</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p2">(<span class="default"><font color="Maroon">sb</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p2">(<span class="default"><font color="Maroon">sc</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T19" target="_self">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p2">(<span class="default"><font color="Maroon">sb</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sc</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sb</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sc</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>thus </b><font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> = 
 <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>
<b>by </b><i>, , , , </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sb</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sc</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sb</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sc</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">sc</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, , , </i>
;<br/>


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