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



<b>assume </b><b>that </b><br/><a NAME="E1:121"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E2:121"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">u</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">t</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:121"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="axioms.html#T20">AXIOMS:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:121"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>
<i><font color="Green">E36</font></i>: <font color="Maroon">r</font> = 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">t</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T74">XCMPLX_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span>

;<br/>
<b>consider </b><font color="Maroon">n</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="E9:121"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<a NAME="E10:121"/><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E11:121"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">u</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">u</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T200">REAL_2:200</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E12:121"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E13:121"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:121_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font>
 ;<br/>

<i><font color="Green">E62</font></i>: 2 <a href="prepower.html#K3" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span>

;<br/>
<a NAME="E3:121_2"/>
2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E4:121_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:121_2"/>
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:121_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="prepower.html#E32"><i><font color="Green">Lemma108</font></i></a>, <a class="ref" href="prepower.html#T28" target="_self">Th28</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:121"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E15:121"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E16:121"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:121"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">u</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E18:121"/>
<font color="Maroon">u</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="txt" href="prepower.html#E32"><i><font color="Green">Lemma108</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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