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<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> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  holds <br/><font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> ) );<br/>
<a NAME="E1:112"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E1:112_1"/>
( 1 <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  )
 <b>by </b><i><a class="ref" href="newton.html#T19">NEWTON:19</a>, <a class="ref" href="newton.html#T20">NEWTON:20</a></i>;<br/>

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

<b>assume </b><a NAME="E2:112_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  )
 ;<br/>

<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:112_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><a NAME="E2:112_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:112_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:112_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T19">NEWTON:19</a></i>;<br/>
<a NAME="E5:112_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E6:112_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E7:112_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:112"/><i><font color="Green">E52</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="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <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="112_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><b>that </b><br/><a NAME="E1:112_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:112_2"/><i><font color="Green">E59</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  holds <br/><font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> ) )
 ;<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="E4:112_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  )
 <b>and </b><br/><a NAME="E5:112_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  holds <br/><font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:112_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 
<b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_2_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_1"><b>suppose </b></a><a NAME="E1:112_2_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">n</font>
;<br/><a NAME="E2:112_2_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</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#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:112_2_1_1"/>
<font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E4:112_2_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E5:112_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E6:112_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:112_2_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><b>thus </b><a NAME="E2:112_2_1_1_1"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_2_1_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:112_2_1_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:112_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:112_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:112_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:112_2_1_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:112_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:112_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="newton.html#K12">!</a> 
 <b>by </b><i/>;<br/><a NAME="E4:112_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:112_2_1_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E6:112_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:112_2_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_2"><b>suppose </b></a><a NAME="E1:112_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">N</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>thus </b><a NAME="E2:112_2_1_2"/>
<font color="Maroon">N</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:112_2_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:112_2_1_2"/>
<font color="Maroon">N</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/><a NAME="E5:112_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E6:112_2_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i>, <a class="ref" href="newton.html#T21">NEWTON:21</a></i>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E7:112_2_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E8:112_2_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:112_2_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 ;<br/><b>thus </b><a NAME="E2:112_2_1_2_1"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_2_1_2_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">N</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:112_2_1_2_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">N</font>
 ;<br/><div class="add"><a NAME="E2:112_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:112_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i/>;<br/><a NAME="E4:112_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E6:112_2_1_2_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:112_2_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:112_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:112_2_1_2_1_1_1_1"/>
<font color="Maroon">N</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/><a NAME="E8:112_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E9:112_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:112_2_1_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="newton.html#T21">NEWTON:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:112_2_1_2_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">N</font>
 ;<br/><div class="add"><a NAME="E2:112_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:112_2_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:112_2_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:112"/>
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="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<b>hence </b><a NAME="E4:112"/>
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="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  holds <br/><font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> ) )
 ;<br/>


</div>
