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

<b>let </b><font color="Maroon">R</font> be  <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:55"/><i><font color="Green">E31</font></i>: 
(  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 2 &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55"/><i><font color="Green">E34</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> &amp; ( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E4:55"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E6:55"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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="E1:55_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:55_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:55_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>set </b><font color="Maroon">H1</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">H2</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font>;<br/>
<a NAME="E8:55"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_3"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E2:55_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font></span>)</span> 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/>
<b>hence </b><a NAME="E3:55_3"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H1</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A</font> as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E10:55"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E11:55"/><b>then </b><i><font color="Green">E41</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> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> is <a href="realset1.html#V1">trivial</a> )
 
;<br/>
<a NAME="E12:55"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4"><b>proof </b></a><div class="add">

<a NAME="E1:55_4"/>
the <a href="struct_0.html#U1">carrier</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">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E4:55"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E2:55_4"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <b>by </b><i>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:55"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/>
<a NAME="E14:55"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font> is <a href="struct_0.html#V3">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_5"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E2:55_5"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font></span>)</span> 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/>
<b>hence </b><a NAME="E3:55_5"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H2</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font> as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E16:55"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E17:55"/><b>then </b><i><font color="Green">E60</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> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> is <a href="realset1.html#V1">trivial</a> )
 
;<br/>
<a NAME="E18:55"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_6"><b>proof </b></a><div class="add">

<a NAME="E1:55_6"/>
the <a href="struct_0.html#U1">carrier</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">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E6:55"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E2:55_6"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <b>by </b><i>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:55"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 
<b>by </b><i><a class="txt" href="neckla_3.html#E11:55"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/>
<a NAME="E20:55"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_7"/>
not <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:55_7"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
<b>hence </b><a NAME="E3:55_7"/>
contradiction
 <b>by </b><i>, <a class="ref" href="card_1.html#T50">CARD_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:55"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/>
<a NAME="E22:55"/><b>then </b><i><font color="Green">E63</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E23:55"/><i><font color="Green">E64</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</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><a class="txt" href="neckla_3.html#E3:55"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/>
<a NAME="E24:55"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E25:55"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/>
<a NAME="E26:55"/><b>then </b><i><font color="Green">E70</font></i>: 
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E27:55"/><i><font color="Green">E92</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_8"/><i><font color="Green">E93</font></i>: 
not the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:55_8"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:55_8"/><b>then </b>
<font color="Maroon">a</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="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:55_8"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</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 <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D6">NECKLACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:55_8"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:55"/>
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"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E29:55"/><b>then </b>
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"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/>
<a NAME="E30:55"/><b>then </b><i><font color="Green">E94</font></i>: 
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E31:55"/><i><font color="Green">E104</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_9"/><i><font color="Green">E105</font></i>: 
not the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:55_9"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:55_9"/><b>then </b>
<font color="Maroon">b</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="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:55_9"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</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 <font color="Maroon">H2</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D6">NECKLACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:55_9"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_10"/>
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E3:55"><i><font color="Green">E34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_10_1"><b>suppose </b></a><a NAME="E1:55_10_1"/><i><font color="Green">E106</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">S</font> =  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font>;<br/><a NAME="E2:55_10_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</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"><font color="Maroon">A</font>,<font color="Maroon">B</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"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/><a NAME="E3:55_10_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><a NAME="E4:55_10_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><a NAME="E5:55_10_1"/><b>then </b><i><font color="Green">E107</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/><a NAME="E6:55_10_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/><b>hence </b><a NAME="E7:55_10_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E10:55"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="neckla_3.html#E13:55"><i><font color="Green">E42</font></i></a>, <a class="txt" href="neckla_3.html#E16:55"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_10_2"><b>suppose </b></a><a NAME="E1:55_10_2"/><i><font color="Green">E108</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">U</font> =  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font>;<br/><a NAME="E2:55_10_2"/><i><font color="Green">E109</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</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="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/><a NAME="E3:55_10_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:55_10_2"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:55"><i><font color="Green">E39</font></i></a>, <a class="txt" href="neckla_3.html#E13:55"><i><font color="Green">E42</font></i></a>, <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, , , , , <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
