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<div class="add">

<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">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:13"/><i><font color="Green">E31</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<i><font color="Green">E32</font></i>: <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i/>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i/>
;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E5:13"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">m1</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m1</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:13_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m1</font>
 ;<br/>

<a NAME="E2:13_3"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:13_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/>
<a NAME="E4:13_3"/><b>then </b>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E5:13_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:13"/>
for <font color="Olive">m1</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m1</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E7:13"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 ;<br/>


</div>
