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

<b>let </b><font color="Maroon" title="c1">Ne</font>, <font color="Maroon" title="c2">Ke</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">Ne</font>, <font color="Olive" title="b2">Ke</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">Ne</font>,<font color="Olive" title="b2">Ke</font>  st <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> holds <br/> ex <font color="Olive" title="b4">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">F</font> ex <font color="Olive" title="b5">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">Ne</font>,<font color="Olive" title="b2">Ke</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b5">G</font> &amp; ( for <font color="Olive" title="b6">i</font>, <font color="Olive" title="b7">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b6">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b7">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) );<br/>
<a NAME="E1:96"/><i><font color="Green" title="E63">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">Ne</font>, <font color="Maroon" title="c4">Me</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">Ne</font>,<font color="Maroon" title="c4">Me</font>;<br/>



<b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E64">A2</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">R</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_1"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">P</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E3:96_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">P</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">R</font>,<font color="Maroon" title="c6">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<a NAME="E5:96_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:96_1"/><b>then </b>
( <font color="Maroon" title="c8">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c8">P</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">P</font> = <font color="Maroon" title="c8">P</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">P</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">F</font>
;<br/>

<a NAME="E8:96_1"/><i><font color="Green" title="E65">A3</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:96_1_1"/><i><font color="Green" title="E66">A4</font></i>: 
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">j</font> )
 ;<br/>

<a NAME="E2:96_1_1"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_1_1"><i><font color="Green" title="E66">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:96_1_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:96_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font>
 
;<br/>
<a NAME="E10:96_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>hence </b><a NAME="E11:96_1"/>
( <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E8:96_1"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> $2</span>)</span></span>)</span>;<br/>
<a NAME="E2:96"/><i><font color="Green" title="E64">A5</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">F</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">F</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F'</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E1:96_2"/><i><font color="Green" title="E65">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F'</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">F'</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">F'</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F'</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>]
 ;<br/>

<b>let </b><font color="Maroon" title="c4">N</font>, <font color="Maroon" title="c5">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c5">K</font>;<br/>



<b>assume </b><a NAME="E2:96_2"/><i><font color="Green" title="E66">A7</font></i>: 
( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F'</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_1"><b>suppose </b></a><a NAME="E1:96_2_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E3:96_2_1_1_1"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> ex <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c5">K</font> st <br/>( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96_2"><i><font color="Green" title="E66">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2"><b>suppose </b></a><a NAME="E1:96_2_1_1_2"/><i><font color="Green" title="E67">A8</font></i>: 
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c5">K</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E2:96_2"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">domF</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E2:96_2"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c9">m</font> =  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font>;<br/><b>set </b><font color="Maroon" title="c10">D</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>;<br/><a NAME="E4:96_2_1_1_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">domF</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/><a NAME="E5:96_2_1_1_2"/><b>then </b><i><font color="Green" title="E68">A9</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_1_1_2_1_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E69">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T66" title="ZFMISC_1:th.66">ZFMISC_1:66</a></i>;<br/><b>set </b><font color="Maroon" title="c11">P</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>;<br/><a NAME="E4:96_2_1_1_2_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/><a NAME="E5:96_2_1_1_2_1_1_1"/><b>then </b>
(  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">P</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E7:96_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c6">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>, <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><a NAME="E8:96_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E70">A11</font></i>: 
<font color="Maroon" title="c12">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/><a NAME="E9:96_2_1_1_2_1_1_1"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E71">A12</font></i>: 
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">j</font> )
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A10</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_1_1"><i><font color="Green" title="E71">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:96_2_1_1_2_1_1_1_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_1_1"><i><font color="Green" title="E71">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:96_2_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> ex <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c5">K</font> st <br/>( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E8:96_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E69">A13</font></i>: 
not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">rFD</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E2:96_2"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A13</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">rF</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E2:96_2"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">dFD</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">FD</font> = <font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">dFD</font>,<font color="Maroon" title="c11">rFD</font> <b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><a NAME="E7:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E70">A14</font></i>: 
<font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_2_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">FD</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E2:96_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c15">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font>, <font color="Maroon" title="c16">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">dFD</font>,<font color="Maroon" title="c11">rFD</font><b> such that </b><br/><a NAME="E9:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E71">A15</font></i>: 
( <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G</font> )
 <b>and </b><br/><a NAME="E10:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E72">A16</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96_2"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">Orde</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">rF</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font><b> such that </b><br/><a NAME="E12:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E73">A17</font></i>: 
<font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>and </b><br/><a NAME="E13:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E74">A18</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b2">k</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T69" target="_self" title="STIRL2_1:th.69">Th69</a></i>;<br/><a NAME="E14:96_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c17">Orde</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E15:96_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c17">Orde</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">Orde'</font> = <font color="Maroon" title="c17">Orde</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font>,<font color="Maroon" title="c12">rF</font> <b>by </b><i><a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">P1</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c12">rF</font><b> such that </b><br/><a NAME="E18:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E75">A19</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> implies <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> implies <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> implies <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="txt" href="stirl2_1.html#E13:96_2_1_1_2_1_1_2"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="stirl2_1.html#E79"><i><font color="Green" title="E59">Lm4</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span>;<br/><a NAME="E19:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E76">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a></i>;<br/><a NAME="E20:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E77">A21</font></i>: 
( <font color="Maroon" title="c13">dFD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c11">rFD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">K</font> )
 ;<br/><a NAME="E21:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E78">A22</font></i>: 
( <font color="Maroon" title="c11">rFD</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon" title="c13">dFD</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/><b>consider </b><font color="Maroon" title="c20">G2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c7">K</font><b> such that </b><br/><a NAME="E23:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E79">A23</font></i>: 
( <font color="Maroon" title="c20">G2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">G</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font> holds <br/><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S2" target="_self" title="STIRL2_1:sch.2">STIRL2_1:sch 2</a>(<a class="txt" href="stirl2_1.html#E19:96_2_1_1_2_1_1_2"><i><font color="Green" title="E76">A20</font></i></a>, <a class="txt" href="stirl2_1.html#E20:96_2_1_1_2_1_1_2"><i><font color="Green" title="E77">A21</font></i></a>, <a class="txt" href="stirl2_1.html#E21:96_2_1_1_2_1_1_2"><i><font color="Green" title="E78">A22</font></i></a>);<br/></i><a NAME="E24:96_2_1_1_2_1_1_2"/>
not <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E25:96_2_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">P2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>,<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span><b> such that </b><br/><a NAME="E27:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E80">A24</font></i>: 
( <font color="Maroon" title="c21">P2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font> &amp; <font color="Maroon" title="c21">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T67" target="_self" title="STIRL2_1:th.67">Th67</a></i>;<br/><a NAME="E28:96_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c15">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c15">P</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; not <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E29:96_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c21">P2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c21">P2</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E27:96_2_1_1_2_1_1_2"><i><font color="Green" title="E80">A24</font></i></a>, <a class="ref" href="stirl2_1.html#T68" target="_self" title="STIRL2_1:th.68">Th68</a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">P2</font> = <font color="Maroon" title="c21">P2</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E31:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E81">A25</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">Gx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_2"/><i><font color="Green" title="E82">A26</font></i>: 
<font color="Maroon" title="c23">Gx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:96_2_1_1_2_1_1_2_2"/><i><font color="Green" title="E83">A27</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Gx</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_2"><i><font color="Green" title="E82">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">dFD</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E6:96_2_1_1_2_1_1_2_2"/><b>then </b>
( (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">dFD</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">dFD</font> ) or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_2"><i><font color="Green" title="E83">A27</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:96_2_1_1_2_1_1_2_2"/><b>then </b>
( ( <font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> &amp; <font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) or ( <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">domF</font> ) )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:96_2_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c23">Gx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_2"><i><font color="Green" title="E83">A27</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:96_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">Fx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_3"/><i><font color="Green" title="E82">A28</font></i>: 
<font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a></i>;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_3"/><b>then </b><i><font color="Green" title="E83">A29</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_1_1_2_1_1_2_3_1_1"/>
( <font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_3"><i><font color="Green" title="E82">A28</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_3"><i><font color="Green" title="E83">A29</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">dFD</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/><b>hence </b><a NAME="E3:96_2_1_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_3_1_1_2"/><i><font color="Green" title="E84">A30</font></i>: 
<font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_3_1_1_2"/>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; not <font color="Maroon" title="c7">K</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:96_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b>
( not  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:96_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font> &amp; <font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:96_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_3_1_1_2"><i><font color="Green" title="E84">A30</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:96_2_1_1_2_1_1_2_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Fx</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:96_2_1_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:96_2_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c23">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E33:96_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E82">A31</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E31:96_2_1_1_2_1_1_2"><i><font color="Green" title="E81">A25</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E34:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E83">A32</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:96_2_1_1_2_1_1_2_4"/>
not <font color="Maroon" title="c7">K</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E2:96_2_1_1_2_1_1_2_4"/><b>then </b><i><font color="Green" title="E84">A33</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_4"/><i><font color="Green" title="E85">A34</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:96_2_1_1_2_1_1_2_4_1"/>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> implies ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_4_1_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_4_1_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96_2_1_1_2_1_1_2_4"><i><font color="Green" title="E84">A33</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:96_2_1_1_2_1_1_2_4_1_1"/>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P2</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E33:96_2_1_1_2_1_1_2"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:96_2_1_1_2_1_1_2_4_1"/>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">P2</font> implies <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:96_2_1_1_2_1_1_2_4"><i><font color="Green" title="E84">A33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:96_2_1_1_2_1_1_2_4"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_4_2"/><i><font color="Green" title="E86">A35</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_1_1_2_1_1_2_4_2_1_1"/>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">dFD</font> )
 </span><b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_4_2"><i><font color="Green" title="E86">A35</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_4_2_1_1_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E87">A36</font></i>: 
( <font color="Maroon" title="c22">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:96_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="txt" href="stirl2_1.html#E27:96_2_1_1_2_1_1_2"><i><font color="Green" title="E80">A24</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96_2_1_1_2_1_1_2_4"><i><font color="Green" title="E84">A33</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_4_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:96_2_1_1_2_1_1_2_4_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:96_2_1_1_2_1_1_2_4_2_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:96_2_1_1_2_1_1_2_4_2_1_1_1"><i><font color="Green" title="E87">A36</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_4_2_1_1_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">dFD</font>
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_4_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">FD</font> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_4_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c14">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> &amp; <font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E4:96_2_1_1_2_1_1_2_4_2_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E27:96_2_1_1_2_1_1_2"><i><font color="Green" title="E80">A24</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:96_2_1_1_2_1_1_2_4_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:96_2_1_1_2_1_1_2_4"/>
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_4"><i><font color="Green" title="E85">A34</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E84">A37</font></i>: 
<font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:96_2_1_1_2_1_1_2_5"/>
<font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_5_1"/><i><font color="Green" title="E85">A38</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_5_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_5_1_1"/>
not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_5_1_1"/><b>then </b>
( ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> &amp; not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> ) or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">dFD</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_5_1"><i><font color="Green" title="E85">A38</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_5_1_1"/><b>then </b>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_5_1_1"/><b>then </b>
( <font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Maroon" title="c16">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_5_1_1"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a></i>;<br/>
<a NAME="E6:96_2_1_1_2_1_1_2_5_1_1"/><b>then </b>
not <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:96_2_1_1_2_1_1_2_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_5_1"><i><font color="Green" title="E85">A38</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:96_2_1_1_2_1_1_2_5_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E7:96_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_5_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:96_2_1_1_2_1_1_2_5_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:96_2_1_1_2_1_1_2_5"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_5_2"/><i><font color="Green" title="E85">A39</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_5_2"/>
( not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_5_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_5_2"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">dFD</font> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:96_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_5_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_5_2"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_5_2"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:96_2_1_1_2_1_1_2_5_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E36:96_2_1_1_2_1_1_2"/><i><font color="Green" title="E85">A40</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> holds <br/><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E86">A41</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_6"/>
not <font color="Maroon" title="c7">K</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_6"/><b>then </b>
( <font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:96_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_6"/><b>then </b>
( <font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> &amp; not <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E35:96_2_1_1_2_1_1_2"><i><font color="Green" title="E84">A37</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_6"><i><font color="Green" title="E86">A41</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_6"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E23:96_2_1_1_2_1_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:96_2_1_1_2_1_1_2_6"/>
<font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c23">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c23">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c23">P21</font> = <font color="Maroon" title="c22">P2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">rF</font>,<font color="Maroon" title="c12">rF</font> <b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/><a NAME="E38:96_2_1_1_2_1_1_2"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E39:96_2_1_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c23">P21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T20" title="FUNCT_2:th.20">FUNCT_2:20</a></i>;<br/><a NAME="E40:96_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">P21</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c23">P21</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">P21</font> = <font color="Maroon" title="c23">P21</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c12">rF</font> <b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E42:96_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E43:96_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E33:96_2_1_1_2_1_1_2"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E44:96_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E86">A42</font></i>: 
( <font color="Maroon" title="c19">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">K</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a>, <a class="ref" href="funct_2.html#T20" title="FUNCT_2:th.20">FUNCT_2:20</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">PG</font> = <font color="Maroon" title="c19">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c7">K</font> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/><a NAME="E46:96_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E47:96_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c20">G2</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E33:96_2_1_1_2_1_1_2"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E48:96_2_1_1_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c12">rF</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/><a NAME="E49:96_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/><a NAME="E50:96_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E87">A43</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">P21</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">PG</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E34:96_2_1_1_2_1_1_2"><i><font color="Green" title="E83">A32</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/><a NAME="E51:96_2_1_1_2_1_1_2"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">PG</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">PG</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">i</font>, <font color="Maroon" title="c27">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E88">A44</font></i>: 
( <font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">PG</font> &amp; <font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">PG</font> &amp; <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">j</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">i1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:96_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E89">A45</font></i>: 
( <font color="Maroon" title="c28">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c28">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="ref" href="stirl2_1.html#T71" target="_self" title="STIRL2_1:th.71">Th71</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">j1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:96_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E90">A46</font></i>: 
( <font color="Maroon" title="c29">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c29">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c29">j1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c20">G2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c29">j1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="ref" href="stirl2_1.html#T71" target="_self" title="STIRL2_1:th.71">Th71</a></i>;<br/>
<a NAME="E6:96_2_1_1_2_1_1_2_7"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">i1</font> = <font color="Maroon" title="c28">i1</font>, <font color="Maroon" title="c31">j1</font> = <font color="Maroon" title="c29">j1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a></i>;<br/>
<a NAME="E8:96_2_1_1_2_1_1_2_7"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; not <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T64" target="_self" title="STIRL2_1:th.64">Th64</a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E9:96_2_1_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E91">A47</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">G2</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E33:96_2_1_1_2_1_1_2"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<a NAME="E10:96_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E92">A48</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">n</font>, <font color="Maroon" title="c33">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_7_1"/><i><font color="Green" title="E93">A49</font></i>: 
( <font color="Maroon" title="c32">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c33">k</font> )
 ;<br/>

<b>assume </b><a NAME="E2:96_2_1_1_2_1_1_2_7_1"/>
<font color="Maroon" title="c32">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c33">k</font>
 ;<br/>

<a NAME="E3:96_2_1_1_2_1_1_2_7_1"/><b>then </b>
<font color="Maroon" title="c32">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c33">k</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_1"><i><font color="Green" title="E93">A49</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:96_2_1_1_2_1_1_2_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E13:96_2_1_1_2_1_1_2"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_1"><i><font color="Green" title="E93">A49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:96_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E93">A50</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">n</font>, <font color="Maroon" title="c33">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_7_2"/><i><font color="Green" title="E94">A51</font></i>: 
( <font color="Maroon" title="c32">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> )
 ;<br/>

<b>assume </b><a NAME="E2:96_2_1_1_2_1_1_2_7_2"/><i><font color="Green" title="E95">A52</font></i>: 
<font color="Maroon" title="c32">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">k</font>
 ;<br/>

<a NAME="E3:96_2_1_1_2_1_1_2_7_2"/><i><font color="Green" title="E96">A53</font></i>: 
( <font color="Maroon" title="c17">Orde</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_7_2"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Orde'</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">Orde'</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_7_2"/><b>then </b>
( <font color="Maroon" title="c32">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">n</font></span>)</span> &amp; <font color="Maroon" title="c33">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">k</font></span>)</span> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_2"><i><font color="Green" title="E94">A51</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7_2"><i><font color="Green" title="E96">A53</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:96_2_1_1_2_1_1_2_7_2"/>
<font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">k</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E10:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E92">A48</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96_2_1_1_2_1_1_2_7_2"><i><font color="Green" title="E95">A52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:96_2_1_1_2_1_1_2_7"/>
( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:96_2_1_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E94">A54</font></i>: 
( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> &amp; (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:96_2_1_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E95">A55</font></i>: 
( <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">i</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">j</font> &amp; <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:96_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E96">A56</font></i>: 
for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_7_3"/><i><font color="Green" title="E97">A57</font></i>: 
( <font color="Maroon" title="c32">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> &amp; <font color="Maroon" title="c32">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_7_3"/>
( <font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_7_3"/><b>then </b><i><font color="Green" title="E98">A58</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_7_3"/><b>then </b><i><font color="Green" title="E99">A59</font></i>: 
( <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E13:96_2_1_1_2_1_1_2"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_3"><i><font color="Green" title="E97">A57</font></i></a>, <a class="ref" href="stirl2_1.html#T8" target="_self" title="STIRL2_1:th.8">Th8</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_7_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:96_2_1_1_2_1_1_2_7_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:96_2_1_1_2_1_1_2_7_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7_3"><i><font color="Green" title="E98">A58</font></i></a>, <a class="ref" href="stirl2_1.html#T8" target="_self" title="STIRL2_1:th.8">Th8</a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:96_2_1_1_2_1_1_2_7_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c32">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E4:96_2_1_1_2_1_1_2_7_3"><i><font color="Green" title="E99">A59</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1"/>
( ( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) or ( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> ) )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_1"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_1"/><i><font color="Green" title="E97">A60</font></i>: 
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_1"/>
<font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">j1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_1_1"/>
<font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c31">j1</font>
 ;<br/>

<a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c31">j1</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E13:96_2_1_1_2_1_1_2"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a></i>;<br/>
<a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E15:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E96">A56</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_1"><i><font color="Green" title="E97">A60</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:96_2_1_1_2_1_1_2_7_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">i</font> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">j</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E18:96_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="stirl2_1.html#E11:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E93">A50</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_1"><i><font color="Green" title="E97">A60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:96_2_1_1_2_1_1_2_7_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_1"/><b>then </b>
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">j1</font> &amp; ( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; ( <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E91">A47</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_1"><i><font color="Green" title="E97">A60</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_1"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">i1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">i1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="stirl2_1.html#E10:96_2_1_1_2_1_1_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="stirl2_1.html#E36:96_2_1_1_2_1_1_2"><i><font color="Green" title="E85">A40</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a></i>;<br/><b>hence </b><a NAME="E5:96_2_1_1_2_1_1_2_7_4_1_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_2"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_2"/><i><font color="Green" title="E97">A61</font></i>: 
( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b>
( ( <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">domF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E91">A47</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b><i><font color="Green" title="E98">A62</font></i>: 
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Maroon" title="c13">dFD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:96_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="stirl2_1.html#E35:96_2_1_1_2_1_1_2"><i><font color="Green" title="E84">A37</font></i></a>, <a class="txt" href="stirl2_1.html#E36:96_2_1_1_2_1_1_2"><i><font color="Green" title="E85">A40</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_2"><i><font color="Green" title="E97">A61</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon" title="c32">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:96_2_1_1_2_1_1_2_7_4_1_2"/><i><font color="Green" title="E99">A63</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">G2</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">j1</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:96_2_1_1_2_1_1_2_7_4_1_2"/>
<font color="Maroon" title="c20">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7_4_1_2"><i><font color="Green" title="E99">A63</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b>
<font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7_4_1_2"><i><font color="Green" title="E99">A63</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/><a NAME="E9:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; not  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7_4_1_2"><i><font color="Green" title="E98">A62</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/><a NAME="E11:96_2_1_1_2_1_1_2_7_4_1_2"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7_4_1_2"><i><font color="Green" title="E98">A62</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E12:96_2_1_1_2_1_1_2_7_4_1_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_3"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_3"/><i><font color="Green" title="E97">A64</font></i>: 
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_3"/><b>then </b>
( ( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E91">A47</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_3"/><b>then </b>
( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E15:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E96">A56</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_3"><i><font color="Green" title="E97">A64</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/><a NAME="E5:96_2_1_1_2_1_1_2_7_4_1_3"/><b>then </b>
( <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c30">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">i</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E18:96_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="stirl2_1.html#E11:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E93">A50</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_3"><i><font color="Green" title="E97">A64</font></i></a></i>;<br/><b>hence </b><a NAME="E6:96_2_1_1_2_1_1_2_7_4_1_3"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="txt" href="stirl2_1.html#E13:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E94">A54</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_4"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_4"/>
( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_4"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_5"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_5"/>
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_5"/><b>then </b>
( <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">i1</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">j1</font> &amp; ( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; ( <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E18:96_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E91">A47</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_5"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">i1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">i1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="stirl2_1.html#E10:96_2_1_1_2_1_1_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="stirl2_1.html#E36:96_2_1_1_2_1_1_2"><i><font color="Green" title="E85">A40</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a></i>;<br/><b>hence </b><a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_5"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_6"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_6"/>
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_6"/><b>then </b><i><font color="Green" title="E97">A65</font></i>: 
( <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">i1</font> &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E18:96_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_6"/>
( <font color="Maroon" title="c17">Orde</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_6"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> &amp; <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c32">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:96_2_1_1_2_1_1_2_7_4_1_6"/><i><font color="Green" title="E98">A66</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Orde'</font> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">i1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:96_2_1_1_2_1_1_2_7_4_1_6"/>
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E6:96_2_1_1_2_1_1_2_7_4_1_6"><i><font color="Green" title="E98">A66</font></i></a>, <a class="ref" href="stirl2_1.html#T8" target="_self" title="STIRL2_1:th.8">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c33">x</font> = <font color="Maroon" title="c32">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E9:96_2_1_1_2_1_1_2_7_4_1_6"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">Orde</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E10:96_2_1_1_2_1_1_2_7_4_1_6"/><b>then </b>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E11:96_2_1_1_2_1_1_2_7_4_1_6"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96_2_1_1_2_1_1_2_7_4_1_6"><i><font color="Green" title="E97">A65</font></i></a>, <a class="txt" href="stirl2_1.html#E6:96_2_1_1_2_1_1_2_7_4_1_6"><i><font color="Green" title="E98">A66</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/><a NAME="E12:96_2_1_1_2_1_1_2_7_4_1_6"/><b>then </b>
( <font color="Maroon" title="c33">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E11:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E93">A50</font></i></a></i>;<br/><b>hence </b><a NAME="E13:96_2_1_1_2_1_1_2_7_4_1_6"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_7"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_7"/>
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_7"/><b>then </b>
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">j1</font> &amp; ( <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; ( <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">FD</font> or <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; not <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E91">A47</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_7"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">i1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">i1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c16">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">j1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:96_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="stirl2_1.html#E10:96_2_1_1_2_1_1_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="stirl2_1.html#E36:96_2_1_1_2_1_1_2"><i><font color="Green" title="E85">A40</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_7"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1_2_1_1_2_7_4_1_8"><b>suppose </b></a><a NAME="E1:96_2_1_1_2_1_1_2_7_4_1_8"/><i><font color="Green" title="E97">A67</font></i>: 
( <font color="Maroon" title="c30">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:96_2_1_1_2_1_1_2_7_4_1_8"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="txt" href="stirl2_1.html#E15:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E96">A56</font></i></a></i>;<br/><a NAME="E3:96_2_1_1_2_1_1_2_7_4_1_8"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">Orde</font> &amp; <font color="Maroon" title="c17">Orde</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c31">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rF</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E5:96_2_1_1_2"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="stirl2_1.html#E12:96_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="txt" href="stirl2_1.html#E5:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E90">A46</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:96_2_1_1_2_1_1_2_7_4_1_8"/><b>then </b>
( ( <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c18">Orde'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Orde</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c31">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E18:96_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="stirl2_1.html#E11:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E93">A50</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_8"><i><font color="Green" title="E97">A67</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/><a NAME="E5:96_2_1_1_2_1_1_2_7_4_1_8"/><b>then </b>
( <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">domF</font></span>)</span> &amp; <font color="Maroon" title="c19">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">i1</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E18:96_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="stirl2_1.html#E3:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E89">A45</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_8"><i><font color="Green" title="E97">A67</font></i></a></i>;<br/><b>hence </b><a NAME="E6:96_2_1_1_2_1_1_2_7_4_1_8"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E88">A44</font></i></a>, <a class="txt" href="stirl2_1.html#E14:96_2_1_1_2_1_1_2_7"><i><font color="Green" title="E95">A55</font></i></a>, <a class="txt" href="stirl2_1.html#E1:96_2_1_1_2_1_1_2_7_4_1_8"><i><font color="Green" title="E97">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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="E17:96_2_1_1_2_1_1_2_7"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">PG</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c27">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E52:96_2_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> ex <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c5">K</font> st <br/>( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E44:96_2_1_1_2_1_1_2"><i><font color="Green" title="E86">A42</font></i></a>, <a class="txt" href="stirl2_1.html#E50:96_2_1_1_2_1_1_2"><i><font color="Green" title="E87">A43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:96_2_1_1_2"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> ex <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c5">K</font> st <br/>( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:96_2"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> ex <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c5">K</font> st <br/>( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:96"/><i><font color="Green" title="E65">A68</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">F</font>]
 
<b>from </b><i><a class="ref" href="stirl2_1.html#S9" target="_self" title="STIRL2_1:sch.9">STIRL2_1:sch 9</a>(<a class="txt" href="stirl2_1.html#E1:96"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:96"><i><font color="Green" title="E64">A5</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Ne</font>,<font color="Maroon" title="c2">Ke</font>;<br/>

<b>assume </b><a NAME="E4:96"/><i><font color="Green" title="E66">A69</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c5">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Ne</font>,<font color="Maroon" title="c2">Ke</font><b> such that </b><br/><a NAME="E6:96"/><i><font color="Green" title="E67">A70</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">G</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">G</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:96"><i><font color="Green" title="E65">A68</font></i></a>, <a class="txt" href="stirl2_1.html#E4:96"><i><font color="Green" title="E66">A69</font></i></a></i>;<br/>
<a NAME="E7:96"/>
<font color="Maroon" title="c5">G</font> is <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E6:96"><i><font color="Green" title="E67">A70</font></i></a>, <a class="ref" href="stirl2_1.html#D5" target="_self" title="STIRL2_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E8:96"/>
 ex <font color="Olive" title="b1">I</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Ne</font>,<font color="Maroon" title="c2">Ke</font> ex <font color="Olive" title="b2">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font> st <br/>( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b1">I</font> is <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E6:96"><i><font color="Green" title="E67">A70</font></i></a></i>;<br/>


</div>
