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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:81"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="orders_1.html#M1">Choice_Function</a> of  <a href="orders_1.html#K1">BOOL</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="orders_2.html#T89" target="_self">Th89</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:81"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E6:81"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:81_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E5:81"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:81_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E6:81"><i><font color="Green">E55</font></i></a>, <a class="ref" href="orders_2.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:81"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E9:81"/><b>then </b>
 <a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E10:81"/><b>then </b>
not  <a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:81"/><b>then </b>
 <a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:81"/><b>then </b>
(  <a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T4">ORDERS_1:4</a></i>;<br/>
<a NAME="E13:81"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D1">ORDERS_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" target="_self">UpperCone</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E15:81"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="orders_2.html#E13:81"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E16:81"/><b>then </b><i><font color="Green">E57</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) )
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E18:81"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a>, <a class="ref" href="orders_2.html#D5" target="_self">Def5</a>, <a class="ref" href="orders_2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E19:81"/><b>then </b><i><font color="Green">E58</font></i>: 
( the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>7</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">c<sub>7</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T93">ORDERS_1:93</a>, <a class="ref" href="orders_1.html#T94">ORDERS_1:94</a>, <a class="ref" href="orders_1.html#T95">ORDERS_1:95</a></i>;<br/>
<a NAME="E20:81"/><i><font color="Green">E59</font></i>: 
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a><br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:81_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:81_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="orders_2.html#E1:81_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="orders_2.html#E2:81_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_2_1_1"/>
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="orders_2.html#E1:81_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="orders_2.html#E2:81_2"><i><font color="Green">E61</font></i></a>, <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="81_2_1_1_1"><b>suppose </b></a><a NAME="E1:81_2_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:81_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a></i>;<br/><b>hence </b><a NAME="E3:81_2_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_1_2"><b>suppose </b></a><a NAME="E1:81_2_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:81_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:81_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E16:81"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E4:81_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:81_2_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_1_3"><b>suppose </b></a><a NAME="E1:81_2_1_1_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:81_2_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:81_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E16:81"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E4:81_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:81_2_1_1_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_1_4"><b>suppose </b></a><a NAME="E1:81_2_1_1_4"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:81_2_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:81_2_1_1_4"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:81_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:81"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:81_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_3_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_1"><b>case </b></a><a NAME="E1:81_3_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E2:81_3_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_3_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>assume </b><a NAME="E3:81_3_1_1_1"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:81_3_1_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E6:81_3_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_3_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="orders_2.html#E5:81_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:81_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_2"><b>case </b></a><a NAME="E1:81_3_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E2:81_3_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></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><b>that </b><br/><a NAME="E1:81_3_1_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E2:81_3_1_1_2_1"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:81_3_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:81_3_1_1_2_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E4:81_3_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:81_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="orders_2.html#E1:81_3_1_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="orders_2.html#E2:81_3_1_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:81_3_1_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E3:81_3_1_1_2"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E4:81_3_1_1_2"/><b>then </b>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:81_3_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="orders_2.html#E183"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E5:81_3_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:81_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E2:81_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="orders_2.html#E1:81_3_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:81_3_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E9:81_3_1_1_2"/><i><font color="Green">E66</font></i>: 
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="orders_2.html#E5:81_3_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><a NAME="E10:81_3_1_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="orders_2.html#E9:81_3_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E11:81_3_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E8:81_3_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_3_1_1_2_3_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:81_3_1_1_2_3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_3_1_1_2_3_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E3:81_3_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_2_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:81_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E3:81_3_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:81_3_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="orders_2.html#E8:81_3_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E5:81_3_1_1_2_3_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E2:81_3_1_1_2_3_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="orders_2.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E6:81_3_1_1_2_3_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E16:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="orders_2.html#E2:81_3_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="orders_2.html#E8:81_3_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E7:81_3_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_2_3_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_2_3_1_1_2"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E3:81_3_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_3_1_1_2_3_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:81_3_1_1_2_3_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_3_1_1_2_3_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:81_3_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E4:81_3_1_1_2_3_1_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><b>then </b><span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="orders_2.html#E10:81_3_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="orders_2.html#E3:81_3_1_1_2_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/><b>hence </b><a NAME="E7:81_3_1_1_2_3_1_1"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:81_3_1_1_2_3_1_2"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_3_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1_1_2_3_1_2"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E9:81_3_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:81_3_1_1_2"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:81_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E22:81"/><b>then </b><i><font color="Green">E60</font></i>: 
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E19:81"><i><font color="Green">E58</font></i></a>, <a class="txt" href="orders_2.html#E20:81"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E23:81"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E19:81"><i><font color="Green">E58</font></i></a>, <a class="txt" href="orders_2.html#E20:81"><i><font color="Green">E59</font></i></a>, <a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/>
<a NAME="E24:81"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self">Def11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:81_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_4_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_4_1_1_1"><b>suppose </b></a><a NAME="E1:81_4_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_4_1_1_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_4_1_1_1_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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><b>that </b><br/><a NAME="E1:81_4_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E2:81_4_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:81_4_1_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:81_4_1_1_1_1_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E4:81_4_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:81_4_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:81_4_1_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:81_4_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E3:81_4_1_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:81_4_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T50" target="_self">Th50</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>9</sub></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="E2:81_4_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:81_4_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="orders_2.html#E2:81_4_1_1_1_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E5:81_4_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E16:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="orders_2.html#E2:81_4_1_1_1_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E6:81_4_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E7:81_4_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E3:81_4_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:81_4_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:81_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_4_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_4_1_1_2"><b>suppose </b></a><a NAME="E1:81_4_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:81_4_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:81_4_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:81_4_1_1_2"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_4_1_1_2_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:81_4_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:81_4_1_1_2_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E3:81_4_1_1_2_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_4_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:81_4_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:81_4_1_1_2_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:81_4_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81_4_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:81_4_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:81_4_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="orders_2.html#E3:81_4_1_1_2_1_1"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E6:81_4_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E7:81_4_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:81_4_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E16:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="orders_2.html#E3:81_4_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="orders_2.html#E6:81_4_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/><b>hence </b><a NAME="E9:81_4_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:81_4_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E3:81_4_1_1_2_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:81_4_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/>


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

<a NAME="E2:81_4_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:81_4_1_1_2_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T65" target="_self">Th65</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:81_4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E3:81_4_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="orders_2.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:81_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="orders_2.html#E22:81"><i><font color="Green">E60</font></i></a>, <a class="txt" href="orders_2.html#E24:81"><i><font color="Green">E61</font></i></a>, <a class="ref" href="orders_2.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E27:81"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E28:81"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D17" target="_self">Def17</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E29:81"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E30:81"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E16:81"><i><font color="Green">E57</font></i></a></i>;<br/>


</div>
