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

<b>set </b><font color="Maroon">N4</font> =  <a href="necklace.html#K4">Necklace</a> 4;<br/>
<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:18"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">R</font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="neckla_2.html#D1" target="_self">NECKLA_2:def 1</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp;  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">S</font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4 );<br/>
<b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="classes2.html#K13">FinSETS</a>  <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font>;<br/>
<a NAME="E7:18"/><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K4">card</a> <font color="Maroon">C</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:18_1"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:18"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>consider </b><font color="Maroon">S</font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E4:18_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 <b>and </b><br/><a NAME="E5:18_2"/><i><font color="Green">E40</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E6:18_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S</font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1"/>
( <font color="Maroon">S</font> is  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  or  ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S</font> is  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:18_2_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> ) ) )
 <b>by </b><i>, <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/><a NAME="E4:18_2_1_1"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="necklace.html#T2">NECKLACE:2</a>, <a class="ref" href="necklace.html#T21">NECKLACE:21</a></i>;<br/><a NAME="E5:18_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
0 <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="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E6:18_2_1_1"/><i><font color="Green">E48</font></i>: 
1 <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="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E7:18_2_1_1"/>
( 0 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 or 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E8:18_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, , <a class="ref" href="necklace.html#T26">NECKLACE:26</a></i>;<br/><a NAME="E9:18_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , </i>;<br/><a NAME="E10:18_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E11:18_2_1_1"/>
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E12:18_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/><a NAME="E13:18_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="neckla_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E14:18_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E15:18_2_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">H1</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">H2</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E3:18_2_1_2"/><i><font color="Green">E81</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>and </b><br/><a NAME="E4:18_2_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  )
 <b>and </b><br/><a NAME="E5:18_2_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> or <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> )
 ;<br/><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font>
 ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:18_2_1_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:18_2_1_2_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/><a NAME="E5:18_2_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="necklace.html#T2">NECKLACE:2</a>, <a class="ref" href="necklace.html#T21">NECKLACE:21</a></i>;<br/><a NAME="E6:18_2_1_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( 0 <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="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 1 <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="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 2 <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="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 3 <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="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E7:18_2_1_2_1"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E8:18_2_1_2_1"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E9:18_2_1_2_1"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E10:18_2_1_2_1"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E11:18_2_1_2_1"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E12:18_2_1_2_1"/><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E13:18_2_1_2_1"/><i><font color="Green">E107</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E14:18_2_1_2_1"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E1:18"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E15:18_2_1_2_1"/><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E16:18_2_1_2_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:18_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E17:18_2_1_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:18"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E18:18_2_1_2_1"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E3:18"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E19:18_2_1_2_1"/><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E4:18"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E20:18_2_1_2_1"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E5:18"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E21:18_2_1_2_1"/><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E7:18"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E22:18_2_1_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_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="E1:18_2_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1"/><i><font color="Green">E118</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E3:18_2_1_2_1_1_1"/><i><font color="Green">E119</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E4:18_2_1_2_1_1_1"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E2:18_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E5:18_2_1_2_1_1_1"/><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E4:18_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E6:18_2_1_2_1_1_1"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:18_2_1_2_1_1_1"/><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_2_1_2_1_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E9:18_2_1_2_1_1_1"/><i><font color="Green">E125</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E5:18_2"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E10:18_2_1_2_1_1_1"/><i><font color="Green">E126</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E6:18_2"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E11:18_2_1_2_1_1_1"/><i><font color="Green">E127</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_1_3_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_3_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:18_2_1_2_1_1_1"/><i><font color="Green">E128</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_1_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_1_4_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_4_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_2_1_2_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E14:18_2_1_2_1_1_1"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E15:18_2_1_2_1_1_1"/><i><font color="Green">E134</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E16:18_2_1_2_1_1_1"/><i><font color="Green">E135</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_1_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_1_5_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_5_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:18_2_1_2_1_1_1"/><i><font color="Green">E136</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_1_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_1_6_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_6_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:18_2_1_2_1_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E19:18_2_1_2_1_1_1"/>
not <font color="Maroon">H2</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E20:18_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E21:18_2_1_2_1_1_1"/><i><font color="Green">E139</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E22:18_2_1_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_1_1_1_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="E3:18_2_1_2_1_1_1_7"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_1_1_1_7_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a>, , , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_7_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_1_7_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_7_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_1_7_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_7_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_7_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_1_7_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_7_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_7_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_1_7_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E23:18_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/><a NAME="E24:18_2_1_2_1_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">H1</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_2_1_1_1_8"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>;<br/>

<b>thus </b><a NAME="E1:18_2_1_2_1_1_1_8_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_1_1_1_8_1_1"/><i><font color="Green">E143</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1_8_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_8_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1_8_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_8_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1_8_1_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_8_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1_8_1_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_8_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1_8_1_1_1_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_8_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_1_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_1_8_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_1_8_1_1_1_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_1_8_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:18_2_1_2_1_1_1_8_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_8_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_1_1_1_8_1_2"/><i><font color="Green">E144</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/>

<a NAME="E2:18_2_1_2_1_1_1_8_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_2_1_2_1_1_1_8_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_2_1_2_1_1_1_8_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_1_8"/>
<font color="Maroon">H1</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i><a class="ref" href="neckla_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:18_2_1_2_1_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">H1</font> is  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>set </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>set </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>;<br/><b>set </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>;<br/><a NAME="E26:18_2_1_2_1_1_1"/>
<font color="Maroon">H1</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E28:18_2_1_2_1_1_1"/>
<font color="Maroon">H2</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E30:18_2_1_2_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E32:18_2_1_2_1_1_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">cH1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_1_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_1_1_1_9"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cS</font>
 ;<br/>

<a NAME="E2:18_2_1_2_1_1_1_9"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:18_2_1_2_1_1_1_9"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:18_2_1_2_1_1_1_9"/>
<font color="Maroon">cH1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:18_2_1_2_1_1_1_9"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:18_2_1_2_1_1_1_9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:18_2_1_2_1_1_1"/>
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E34:18_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">cH1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E35:18_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">cH1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">cS</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E36:18_2_1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E37:18_2_1_2_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2"/><i><font color="Green">E153</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E3:18_2_1_2_1_1_2"/><i><font color="Green">E154</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E4:18_2_1_2_1_1_2"/><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E2:18_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E5:18_2_1_2_1_1_2"/><i><font color="Green">E156</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E4:18_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E6:18_2_1_2_1_1_2"/><i><font color="Green">E157</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:18_2_1_2_1_1_2"/><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_2_1_2_1_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E9:18_2_1_2_1_1_2"/><i><font color="Green">E160</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E5:18_2"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E10:18_2_1_2_1_1_2"/><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E6:18_2"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E11:18_2_1_2_1_1_2"/><i><font color="Green">E162</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_2_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_2_3_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_3_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:18_2_1_2_1_1_2"/><i><font color="Green">E163</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_2_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_2_4_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_4_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_2_1_2_1_1_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E14:18_2_1_2_1_1_2"/><i><font color="Green">E165</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E15:18_2_1_2_1_1_2"/><i><font color="Green">E166</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E16:18_2_1_2_1_1_2"/><i><font color="Green">E167</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_2_5_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_5_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:18_2_1_2_1_1_2"/><i><font color="Green">E168</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_1_1_2_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_1_1_2_6_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_6_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:18_2_1_2_1_1_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E19:18_2_1_2_1_1_2"/>
not <font color="Maroon">H1</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E20:18_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E170</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E21:18_2_1_2_1_1_2"/><i><font color="Green">E171</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E22:18_2_1_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_1_1_2_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="E3:18_2_1_2_1_1_2_7"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_1_1_2_7_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a>, , , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_7_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_2_7_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_7_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_2_7_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_7_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_7_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_2_7_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_7_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_7_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1_2_7_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E23:18_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/><a NAME="E24:18_2_1_2_1_1_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">H2</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_8"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_2_1_1_2_8"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>;<br/>

<b>thus </b><a NAME="E1:18_2_1_2_1_1_2_8_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_1_1_2_8_1_1"/><i><font color="Green">E175</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2_8_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_8_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2_8_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_8_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2_8_1_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_8_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2_8_1_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_8_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2_8_1_1_1_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_8_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_1_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_1_1_2_8_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_1_1_2_8_1_1_1_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_1_2_8_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:18_2_1_2_1_1_2_8_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_8_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_1_1_2_8_1_2"/><i><font color="Green">E176</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/>

<a NAME="E2:18_2_1_2_1_1_2_8_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_2_1_2_1_1_2_8_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_2_1_2_1_1_2_8_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_1_1_2_8"/>
<font color="Maroon">H2</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i><a class="ref" href="neckla_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:18_2_1_2_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">H2</font> is  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>set </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>set </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>;<br/><b>set </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>;<br/><a NAME="E26:18_2_1_2_1_1_2"/>
<font color="Maroon">H1</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E28:18_2_1_2_1_1_2"/>
<font color="Maroon">H2</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E30:18_2_1_2_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E32:18_2_1_2_1_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">cH2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1_1_2_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_1_1_2_9"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cS</font>
 ;<br/>

<a NAME="E2:18_2_1_2_1_1_2_9"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:18_2_1_2_1_1_2_9"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:18_2_1_2_1_1_2_9"/>
<font color="Maroon">cH1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:18_2_1_2_1_1_2_9"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:18_2_1_2_1_1_2_9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:18_2_1_2_1_1_2"/>
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E34:18_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">cH2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E35:18_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">cH2</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">cS</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E36:18_2_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E37:18_2_1_2_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font>
 ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:18_2_1_2_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:18_2_1_2_2"/><i><font color="Green">E184</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/><a NAME="E5:18_2_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="necklace.html#T2">NECKLACE:2</a>, <a class="ref" href="necklace.html#T21">NECKLACE:21</a></i>;<br/><a NAME="E6:18_2_1_2_2"/><b>then </b><i><font color="Green">E185</font></i>: 
( 0 <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="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 1 <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="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 2 <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="necklace.html#K4">Necklace</a> 4</span>)</span> &amp; 3 <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="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>set </b><font color="Maroon">A</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>;<br/><b>set </b><font color="Maroon">B</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>;<br/><b>set </b><font color="Maroon">C</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>set </b><font color="Maroon">D</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>set </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>;<br/><b>set </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>;<br/><b>set </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><a NAME="E7:18_2_1_2_2"/><i><font color="Green">E186</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E8:18_2_1_2_2"/><i><font color="Green">E187</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E9:18_2_1_2_2"/><i><font color="Green">E188</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E10:18_2_1_2_2"/><i><font color="Green">E189</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E11:18_2_1_2_2"/><i><font color="Green">E190</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E12:18_2_1_2_2"/><i><font color="Green">E191</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><a NAME="E13:18_2_1_2_2"/><i><font color="Green">E192</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E14:18_2_1_2_2"/><i><font color="Green">E193</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E1:18"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E15:18_2_1_2_2"/><i><font color="Green">E194</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E16:18_2_1_2_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:18_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E17:18_2_1_2_2"/><b>then </b><i><font color="Green">E195</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:18"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E18:18_2_1_2_2"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E19:18_2_1_2_2"/><i><font color="Green">E197</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E3:18"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E20:18_2_1_2_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E21:18_2_1_2_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:18"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E22:18_2_1_2_2"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E23:18_2_1_2_2"/><i><font color="Green">E199</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E4:18"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E24:18_2_1_2_2"/><i><font color="Green">E200</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E5:18"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E25:18_2_1_2_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E26:18_2_1_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:18"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E27:18_2_1_2_2"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E28:18_2_1_2_2"/><i><font color="Green">E202</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E7:18"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E29:18_2_1_2_2"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E30:18_2_1_2_2"/><i><font color="Green">E204</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E31:18_2_1_2_2"/><i><font color="Green">E205</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E32:18_2_1_2_2"/><i><font color="Green">E206</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_1"/><i><font color="Green">E207</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="ref" href="neckla_2.html#T3" target="_self">Th3</a>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_1_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_1_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_1_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E33:18_2_1_2_2"/><i><font color="Green">E208</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_2"/><i><font color="Green">E209</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_2_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_2_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_2_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_2_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_2_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_2_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_2_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_2_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_2_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E34:18_2_1_2_2"/><i><font color="Green">E210</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_3"/><i><font color="Green">E211</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="ref" href="neckla_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_3_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_3_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_3_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_3_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_3_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_3_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_3_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_3_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,3</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_3_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E35:18_2_1_2_2"/><i><font color="Green">E212</font></i>: 
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_2_2_4"/><i><font color="Green">E213</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_2_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/>

<a NAME="E2:18_2_1_2_2_4_1"/><b>then </b><i><font color="Green">E214</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:18_2_1_2_2_4_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:18_2_1_2_2_4_1"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:18_2_1_2_2_4_1"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:18_2_1_2_2_4_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:18_2_1_2_2_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_2_1_2_2_4"/><i><font color="Green">E216</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_4_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_2_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/>

<a NAME="E2:18_2_1_2_2_4_2"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:18_2_1_2_2_4_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:18_2_1_2_2_4_2"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:18_2_1_2_2_4_2"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:18_2_1_2_2_4_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:18_2_1_2_2_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18_2_1_2_2_4"/><i><font color="Green">E219</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_4_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_2_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:18_2_1_2_2_4_3"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:18_2_1_2_2_4_3"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:18_2_1_2_2_4_3"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:18_2_1_2_2_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_2_1_2_2_4"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_4_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_2_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:18_2_1_2_2_4_4"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:18_2_1_2_2_4_4"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:18_2_1_2_2_4_4"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E5:18_2_1_2_2_4_4"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E6:18_2_1_2_2_4_4"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E7:18_2_1_2_2_4_4"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:18_2_1_2_2_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_2_1_2_2_4"/>
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_5"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:18_2_1_2_2_5_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#T3" target="_self">Th3</a>, <a class="ref" href="neckla_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:18_2_1_2_2_5_1"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_3_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:18_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">x1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x2</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x3</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x4</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x5</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x6</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E5:18_2_1_2_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E2:18_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E6:18_2_1_2_2_5_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E230</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:18_2_1_2_2_5_1"/><i><font color="Green">E231</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_4"><b>proof </b></a><div class="add">

<div><i><font color="Green">E232</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_4_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_4_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E233</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_4_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_4_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_4_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_4_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_4_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:18_2_1_2_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E4:18_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E10:18_2_1_2_2_5_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E234</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:18_2_1_2_2_5_1"/><i><font color="Green">E235</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_5"><b>proof </b></a><div class="add">

<div><i><font color="Green">E236</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_5_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_5_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E237</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_5_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_5_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_5_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_5_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_5_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_2_1_2_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E6:18_2"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E14:18_2_1_2_2_5_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E15:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E238</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E16:18_2_1_2_2_5_1"/><i><font color="Green">E239</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_6"><b>proof </b></a><div class="add">

<div><i><font color="Green">E240</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_6_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_6_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E241</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_6_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_6_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_6_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_6_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_6_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:18_2_1_2_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E18:18_2_1_2_2_5_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E19:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E242</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E20:18_2_1_2_2_5_1"/><i><font color="Green">E243</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_7"><b>proof </b></a><div class="add">

<div><i><font color="Green">E244</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_7_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_7_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_7_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E245</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_7_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_7_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_7_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_7_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_7_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_7_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_7_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_1_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:18_2_1_2_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E22:18_2_1_2_2_5_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E23:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E246</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E24:18_2_1_2_2_5_1"/><i><font color="Green">E247</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_8"><b>proof </b></a><div class="add">

<div><i><font color="Green">E248</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_8_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_8_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_8_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E249</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_8_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_8_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_8_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_8_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_8_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_8_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_1_8"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:18_2_1_2_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E26:18_2_1_2_2_5_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E27:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E250</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E28:18_2_1_2_2_5_1"/><i><font color="Green">E251</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_9"><b>proof </b></a><div class="add">

<div><i><font color="Green">E252</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_9_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_9_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_9_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E253</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_9_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_9_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_9_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_9_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_1_9_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_1_9_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_9_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_1_9"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:18_2_1_2_2_5_1"/>
not <font color="Maroon">H2</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E30:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E254</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E31:18_2_1_2_2_5_1"/><i><font color="Green">E255</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E32:18_2_1_2_2_5_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_2_5_1_10"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="E3:18_2_1_2_2_5_1_10"/><i><font color="Green">E256</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_5_1_10_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, , , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_10_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_10_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_1_10_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_10_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_10_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_1_10_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_10_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_10_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_1_10_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_10_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_10_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_1_10_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E33:18_2_1_2_2_5_1"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/><a NAME="E34:18_2_1_2_2_5_1"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">H1</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_11"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_2_2_5_1_11"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>;<br/>

<b>thus </b><a NAME="E1:18_2_1_2_2_5_1_11_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_5_1_11_1_1"/><i><font color="Green">E259</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1_11_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_11_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1_11_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_11_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1_11_1_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_11_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1_11_1_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_11_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1_11_1_1_1_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_11_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_1_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_1_11_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_1_11_1_1_1_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_1_11_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:18_2_1_2_2_5_1_11_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_11_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_5_1_11_1_2"/><i><font color="Green">E260</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/>

<a NAME="E2:18_2_1_2_2_5_1_11_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_2_1_2_2_5_1_11_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18_2_1_2_2_5_1_11_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E5:18_2_1_2_2_5_1_11_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E6:18_2_1_2_2_5_1_11_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_1_11"/>
<font color="Maroon">H1</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i><a class="ref" href="neckla_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:18_2_1_2_2_5_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">H1</font> is  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E36:18_2_1_2_2_5_1"/>
<font color="Maroon">H1</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E38:18_2_1_2_2_5_1"/>
<font color="Maroon">H2</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E40:18_2_1_2_2_5_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E42:18_2_1_2_2_5_1"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">cH1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_1_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_5_1_12"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cS</font>
 ;<br/>

<a NAME="E2:18_2_1_2_2_5_1_12"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:18_2_1_2_2_5_1_12"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:18_2_1_2_2_5_1_12"/>
<font color="Maroon">cH1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:18_2_1_2_2_5_1_12"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:18_2_1_2_2_5_1_12"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:18_2_1_2_2_5_1"/>
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E44:18_2_1_2_2_5_1"/><b>then </b>
<font color="Maroon">cH1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E45:18_2_1_2_2_5_1"/><b>then </b>
<font color="Maroon">cH1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">cS</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E46:18_2_1_2_2_5_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E47:18_2_1_2_2_5_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:18_2_1_2_2_5_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#T3" target="_self">Th3</a>, <a class="ref" href="neckla_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:18_2_1_2_2_5_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_3_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="neckla_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_2_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:18_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">x1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x2</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x3</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x4</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x5</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">x6</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E5:18_2_1_2_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E2:18_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E6:18_2_1_2_2_5_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E270</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:18_2_1_2_2_5_2"/><i><font color="Green">E271</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_4"><b>proof </b></a><div class="add">

<div><i><font color="Green">E272</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_4_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_4_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E273</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_4_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_4_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_4_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_4_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_4_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:18_2_1_2_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E4:18_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E10:18_2_1_2_2_5_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E274</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:18_2_1_2_2_5_2"/><i><font color="Green">E275</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_5"><b>proof </b></a><div class="add">

<div><i><font color="Green">E276</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_5_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_5_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E277</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_5_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_5_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_5_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_5_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_5_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_2_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_2_1_2_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E6:18_2"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E14:18_2_1_2_2_5_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E15:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E278</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E16:18_2_1_2_2_5_2"/><i><font color="Green">E279</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_6"><b>proof </b></a><div class="add">

<div><i><font color="Green">E280</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_6_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_6_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E281</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_6_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_6_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_6_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_6_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_6_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:18_2_1_2_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E18:18_2_1_2_2_5_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E19:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E282</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E20:18_2_1_2_2_5_2"/><i><font color="Green">E283</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_7"><b>proof </b></a><div class="add">

<div><i><font color="Green">E284</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_7_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_7_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_7_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E285</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_7_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_7_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_7_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_7_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_7_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_7_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_7_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_2_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:18_2_1_2_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E22:18_2_1_2_2_5_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E23:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E286</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E24:18_2_1_2_2_5_2"/><i><font color="Green">E287</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_8"><b>proof </b></a><div class="add">

<div><i><font color="Green">E288</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_8_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_8_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_8_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E289</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_8_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_8_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_8_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_8_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_8_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_8_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_2_8"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:18_2_1_2_2_5_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E26:18_2_1_2_2_5_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E27:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E290</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E28:18_2_1_2_2_5_2"/><i><font color="Green">E291</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_9"><b>proof </b></a><div class="add">

<div><i><font color="Green">E292</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_9_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_9_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_9_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E293</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_9_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_9_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_9_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_9_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_2_2_5_2_9_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:18_2_1_2_2_5_2_9_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_9_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_2_2_5_2_9"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:18_2_1_2_2_5_2"/>
not <font color="Maroon">H1</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E30:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E294</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E31:18_2_1_2_2_5_2"/><i><font color="Green">E295</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E32:18_2_1_2_2_5_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_2_1_2_2_5_2_10"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="E3:18_2_1_2_2_5_2_10"/><i><font color="Green">E296</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_5_2_10_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, , , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_10_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_10_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_2_10_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_10_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_10_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_2_10_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_10_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_10_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_2_10_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_10_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_10_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_2_5_2_10_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E33:18_2_1_2_2_5_2"/><b>then </b><i><font color="Green">E297</font></i>: 
<font color="Maroon">f</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="necklace.html#K4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/><a NAME="E34:18_2_1_2_2_5_2"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">H2</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_11"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_2_2_5_2_11"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>;<br/>

<b>thus </b><a NAME="E1:18_2_1_2_2_5_2_11_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_5_2_11_1_1"/><i><font color="Green">E299</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2_11_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_11_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2_11_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_11_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1_3"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2_11_1_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_11_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1_4"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2_11_1_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_11_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1_5"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2_11_1_1_1_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_11_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_1_1_6"><b>suppose </b></a><a NAME="E1:18_2_1_2_2_5_2_11_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_2_2_5_2_11_1_1_1_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_2_2_5_2_11_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:18_2_1_2_2_5_2_11_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_11_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_5_2_11_1_2"/><i><font color="Green">E300</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font>
 ;<br/>

<a NAME="E2:18_2_1_2_2_5_2_11_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_2_1_2_2_5_2_11_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18_2_1_2_2_5_2_11_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E5:18_2_1_2_2_5_2_11_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E6:18_2_1_2_2_5_2_11_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_2_2_5_2_11"/>
<font color="Maroon">H2</font> <a href="necklace.html#R1">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i><a class="ref" href="neckla_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:18_2_1_2_2_5_2"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">H2</font> is  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E36:18_2_1_2_2_5_2"/>
<font color="Maroon">H1</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E38:18_2_1_2_2_5_2"/>
<font color="Maroon">H2</font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E40:18_2_1_2_2_5_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">cS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E42:18_2_1_2_2_5_2"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">cH2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_2_5_2_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_1_2_2_5_2_12"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cS</font>
 ;<br/>

<a NAME="E2:18_2_1_2_2_5_2_12"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:18_2_1_2_2_5_2_12"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH1</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:18_2_1_2_2_5_2_12"/>
<font color="Maroon">cH1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:18_2_1_2_2_5_2_12"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cH2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:18_2_1_2_2_5_2_12"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:18_2_1_2_2_5_2"/>
<font color="Maroon">cS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cH1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">cH2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E44:18_2_1_2_2_5_2"/><b>then </b>
<font color="Maroon">cH2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E45:18_2_1_2_2_5_2"/><b>then </b>
<font color="Maroon">cH2</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">cS</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E46:18_2_1_2_2_5_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">cS</font>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E47:18_2_1_2_2_5_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:18_2_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:18"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7">NAT_1:sch 7</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">S</font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E11:18"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 <b>and </b><br/><a NAME="E12:18"/><i><font color="Green">E306</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E13:18"/>
<font color="Maroon">S</font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 ;<br/>
<b>thus </b><a NAME="E14:18"/>
contradiction
 <b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>


</div>
