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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>assume </b><a NAME="E1:100"/><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">s</font> being   <a href="prepower.html#NM1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:100"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">s</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">s</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">s</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="E4:100"/><i><font color="Green">E34</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/>
<a NAME="E5:100"/><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">s</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_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="E1:100_1"/>
<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> <a href="hidden.html#R1">=</a> <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/><b>hence </b><a NAME="E2:100_1"/>
<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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100"/>
<font color="Maroon">a</font> <a href="prepower.html#K12" target="_self">#R</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T31">SEQ_2:31</a></i>;<br/>


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