<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<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> ] means for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds <font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K8" target="_self">choose</a> <font color="Olive">b<sub>1</sub></font> is   <a href="nat_1.html#NM1">Nat</a>;<br/>
<a NAME="E1:38"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:38_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<b>hence </b><a NAME="E2:38_1"/>
0 <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="newton.html#D3" target="_self">Def3</a>, <a class="ref" href="newton.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:38"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<a NAME="E2:38_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K8" target="_self">choose</a> <font color="Olive">b<sub>1</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:38_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:38_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:38_2_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="newton.html#E1:38_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K8" target="_self">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="newton.html#E1:38_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:38_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K8" target="_self">choose</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:38_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="newton.html#E2:38_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="newton.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E3:38_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
;<br/>
<b>hence </b><a NAME="E4:38_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="newton.html#E2:38_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="newton.html#T29" target="_self">Th29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:38"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="txt" href="newton.html#E1:38"><i><font color="Green">E23</font></i></a>, <a class="txt" href="newton.html#E2:38"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K8" target="_self">choose</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 ;<br/>


</div>
