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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">A</font> being <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a>   st  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">A</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">a<sub>1</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> ;<br/>
<a NAME="E1:17"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> 0</span>)</span>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a><br/>

<a NAME="E4:17_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
( <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a> &amp;  <a href="yellow_1.html#K2">InclPoset</a> 0 is <a href="struct_0.html#V3">empty</a> )
 
<b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">A</font> being <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a>   st  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">A</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 ;<br/>

<b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a> ;<br/>

<b>assume </b><a NAME="E2:17_2"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E3:17_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:17_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:17_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<a NAME="E6:17_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1"/>
( <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_2_1_1"/><i><font color="Green">E59</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E4:17_2_1_1"/>
<font color="Maroon">A</font>, <a href="yellow_1.html#K2">InclPoset</a> 1 <a href="waybel_1.html#R5">are_isomorphic</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">g</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="pboole.html#K9">--&gt;</a> 0;<br/>
<a NAME="E1:17_2_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E2:17_2_1_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="pboole.html#K9">--&gt;</a> 0 as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> 1</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:17_2_1_1_1"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_2_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_1_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E2:17_2_1_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:17_2_1_1_1_1"/>
for <font color="Olive">e</font>, <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Olive">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">f</font> iff <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_2_1_1_1_1_2_1"/>
<font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:17_2_1_1_1_1_2_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_1_1_2_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:17_2_1_1_1_1_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:17_2_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:17_2_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2_1_1_1_1"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_2_1_1_1"/>
<font color="Maroon">A</font>, <a href="yellow_1.html#K2">InclPoset</a> 1 <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:17_2_1_1"/>
<font color="Maroon">A</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:17_2_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><i><font color="Green">E64</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">n</font>

;<br/><b>then reconsider </b><font color="Maroon">Ab</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Ab</font> as  <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a>  <b>by </b><i/>;<br/><a NAME="E6:17_2_1_2"/>
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E7:17_2_1_2"/><b>then </b>
<font color="Maroon">B</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E9:17_2_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/><a NAME="E10:17_2_1_2"/><i><font color="Green">E66</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E11:17_2_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E12:17_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E13:17_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a>, </i>;<br/><a NAME="E14:17_2_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E15:17_2_1_2"/><i><font color="Green">E69</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="orders_4.html#R1" target="_self">form_upper_lower_partition_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E16:17_2_1_2"/><i><font color="Green">E70</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><i><font color="Green">E71</font></i>: <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 = 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="card_1.html#T52">CARD_1:52</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
;<br/><a NAME="E18:17_2_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E19:17_2_1_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:17_2_1_2_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<a NAME="E2:17_2_1_2_3"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:17_2_1_2_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:17_2_1_2_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E6:17_2_1_2_3"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2_1_2_3_1"/>
not <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_2_1_2_3_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:17_2_1_2_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, , <a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">w'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:17_2_1_2_3_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">w'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> )
 ;<br/>
<a NAME="E7:17_2_1_2_3_1"/>
<font color="Maroon">w'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:17_2_1_2_3_1"/><b>then </b>
<font color="Maroon">w'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E9:17_2_1_2_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_2_1_2_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E8:17_2_1_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:17_2_1_2_3"/>
<font color="Maroon">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:17_2_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> <a href="orders_4.html#R1" target="_self">form_upper_lower_partition_of</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">X</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E21:17_2_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E23:17_2_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">n'</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>set </b><font color="Maroon">X'</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font>;<br/><b>set </b><font color="Maroon">Y'</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font>;<br/><a NAME="E25:17_2_1_2"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>set </b><font color="Maroon">g</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">n</font>;<br/><a NAME="E26:17_2_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">n</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E28:17_2_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> holds  <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E29:17_2_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E30:17_2_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E31:17_2_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_2_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span>
 ;<br/>

<a NAME="E2:17_2_1_2_4"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1_2_4"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E1:17"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:17_2_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span> <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E34:17_2_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T70">PARTFUN1:70</a></i>;<br/><a NAME="E35:17_2_1_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">n'</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T61">FUNCT_2:61</a></i>;<br/><a NAME="E36:17_2_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E37:17_2_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">n'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T62">FUNCT_2:62</a></i>;<br/><a NAME="E38:17_2_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:17_2_1_2_5"/>
for <font color="Olive">e</font>, <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> holds <br/> ( <font color="Olive">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">f</font> iff <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>, <font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>

<a NAME="E1:17_2_1_2_5_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E2:17_2_1_2_5_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:17_2_1_2_5_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E5:17_2_1_2_5_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">b'</font></span>)</span> <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_2_1_2_5_1_1"/>
<font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:17_2_1_2_5_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E3:17_2_1_2_5_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:17_2_1_2_5_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="orders_4.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:17_2_1_2_5_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E9:17_2_1_2_5_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E2:17"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:17_2_1_2_5_1"/>
<font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_1_2_5"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">B2</font> = <font color="Maroon">n'</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E40:17_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E41:17_2_1_2"/><b>then </b>
( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E42:17_2_1_2"/><b>then </b>
(  <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ab</font> &amp; <font color="Maroon">g</font> is   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E44:17_2_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font> &amp; <font color="Maroon">h</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 <b>by </b><i>, , , , </i>;<br/><b>thus </b><a NAME="E45:17_2_1_2"/>
<font color="Maroon">A</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="orders_4.html#E2:17"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:17"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:17"/>
for <font color="Olive">A</font> being <a href="group_1.html#V6">finite</a>  <a href="orders_4.html#M1" target="_self">Chain</a> <br/> for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> holds <br/><font color="Olive">A</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Olive">n</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 ;<br/>


</div>
