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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">p</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:117"/><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">s1</font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:117"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">s1</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">s1</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">s1</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:117"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:117"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <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">s1</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:117"/><b>then </b><i><font color="Green">E36</font></i>: 
 <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">s1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_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="E1:117_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E2:117_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s1</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">s2</font> = <font color="Maroon">p</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s1</font> as   <a href="prepower.html#NM1">Rational_Sequence</a> <b>by </b><i/>;<br/>
<a NAME="E9:117"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<a NAME="E10:117"/><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">s2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:117"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>;<br/>
<a NAME="E12:117"/><b>then </b><i><font color="Green">E55</font></i>: 
<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#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">s2</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_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/><i><font color="Green">E59</font></i>: <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">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> = 
<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">s1</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">p</font>
<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"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="prepower.html#E1:117_2"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><a NAME="E2:117_2"/>
<font color="Maroon">a</font> <a href="prepower.html#K8" target="_self">#Q</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="prepower.html#K10" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:117_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <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">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K11" target="_self">#Q</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:117"/>
<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="prepower.html#K8" target="_self">#Q</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <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#K3">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, , , , , </i>;<br/>


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