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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f</font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:48"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:48_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1"/>
<font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E4:48"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E5:48"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/><font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/>

<b>assume </b><a NAME="E2:48_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:48_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:48_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, , <a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E7:48_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E8:48_2"/><b>then </b>
<font color="Maroon">x</font> <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="E9:48_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E10:48_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E11:48_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:48_2"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:48_2"/>
contradiction
 <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:48"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/>

<b>assume </b><a NAME="E2:48_3"/><i><font color="Green">E76</font></i>: 
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:48_3"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:48_3"/><b>then </b>
( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:48_3"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:48_3"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<a NAME="E7:48_3"/><b>then </b><i><font color="Green">E77</font></i>: 
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:48_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:48_3"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:48_3"/><b>then </b>
<font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:48_3"/>
contradiction
 <b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:48"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_4"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/>

<b>assume </b><a NAME="E2:48_4"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:48_4"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:48_4"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:48"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_5"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_5"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:48_5"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:48_5"/><b>then </b>
<font color="Maroon">k</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E6:48_5"/><b>then </b>
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:48_5"/><b>then </b><i><font color="Green">E106</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E8:48_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:48"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E10:48_5"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E11:48_5"/><b>then </b>
( <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="txt" href="stirl2_1.html#E3:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:48_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:48_5"/><b>then </b><i><font color="Green">E107</font></i>: 
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E14:48_5"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E15:48_5"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E16:48_5"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_5_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:48_5_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:48_5_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E4:48_5_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">n1</font> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_5_1_1_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a>, <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="48_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_5_1_1_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:48_5_1_1_1_1"/><b>then </b>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:48_5_1_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/><a NAME="E4:48_5_1_1_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:48_5_1_1_1_1"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_5_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_5_1_1_1_2"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, <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><b>hence </b><a NAME="E6:48_5_1"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:48_5"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_6_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1_1"><b>suppose </b></a><a NAME="E1:48_6_1_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:48_6_1_1"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_6_1_1"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1_2"><b>suppose </b></a><a NAME="E1:48_6_1_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:48_6_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_6_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E4:48_6_1_2"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="txt" href="stirl2_1.html#E1:48"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E5:48_6_1_2"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a>, <a class="txt" href="stirl2_1.html#E2:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E6:48_6_1_2"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <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="48_6_1_3"><b>suppose </b></a><a NAME="E1:48_6_1_3"/>
( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:48_6_1_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E3:48_6_1_3"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <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="48_6_1_4"><b>suppose </b></a><a NAME="E1:48_6_1_4"/>
( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:48_6_1_4"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:48_6_1_4"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:48"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>


</div>
