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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">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="E1:14"/><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> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green">E35</font></i>: <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span> = 
<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> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</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#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>

;<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</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span> = 
<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> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>

;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:14"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">m1</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">m1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m1</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:14_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/>

<a NAME="E2:14_3"/>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:14_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span></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> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span></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#T99">XREAL_1:99</a></i>;<br/>
<a NAME="E4:14_3"/><b>then </b>
<font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span></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="E5:14_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</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="E9:14"/>
for <font color="Olive">m1</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">m1</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E10:14"/>
<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>
 <b>by </b><i>, </i>;<br/>


</div>
