<?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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <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>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <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>3</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E4:68"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E5:68"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
not  <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>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:68"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E8:68"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:68"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E11:68"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="orders_2.html#E11:68"><i><font color="Green">E52</font></i></a>, <a class="ref" href="orders_2.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E13:68"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:68"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:68"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:68"><i><font color="Green">E50</font></i></a>, <a class="txt" href="orders_2.html#E10:68"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E15:68"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E16:68"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is  not    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T84" target="_self">Th84</a></i>;<br/>
<a NAME="E17:68"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="orders_2.html#E10:68"><i><font color="Green">E51</font></i></a>, <a class="txt" href="orders_2.html#E13:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="orders_2.html#T83" target="_self">Th83</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</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="E19:68"/><i><font color="Green">E54</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="E20:68"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E21:68"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:68"><i><font color="Green">E49</font></i></a>, <a class="txt" href="orders_2.html#E11:68"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E22:68"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>2</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>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">c<sub>2</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>
  <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="68_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="E1:68_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>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:68_1_1"/><b>then </b><i><font color="Green">E57</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>2</sub></font>
 
;<br/>
<a NAME="E3:68_1_1"/><b>then </b><i><font color="Green">E58</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/>
<a NAME="E4:68_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"> <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> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</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="E6:68_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E7:68_1_1"/><i><font color="Green">E60</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> <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="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>
<a NAME="E8:68_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="E9:68_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:68_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E10:68_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:68_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="orders_2.html#E6:68_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:68_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E13:68_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="orders_2.html#E13:68_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="orders_2.html#D17" target="_self">Def17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:68_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">(<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>
 <b>by </b><i><a class="txt" href="orders_2.html#E3:68_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="orders_2.html#E6:68_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="orders_2.html#E12:68_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:68_1_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>8</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>12</sub></font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>7</sub></font> )
 </span><b>by </b><i><a class="txt" href="orders_2.html#E1:68_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="orders_2.html#T83" target="_self">Th83</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_1_2_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="txt" href="orders_2.html#E19:68"><i><font color="Green">E54</font></i></a>, <a class="txt" href="orders_2.html#E6:68_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="orders_2.html#E9:68_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="orders_2.html#E12:68_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="orders_2.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E3:68_1_1_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">(<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>
 <b>by </b><i><a class="txt" href="orders_2.html#E3:68_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:68_1_1_1_1_2_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>8</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T84" target="_self">Th84</a></i>;<br/><a NAME="E3:68_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</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_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:68_1_1_1_1_2_1_1_2"/><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>7</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E3:68_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="orders_2.html#E6:68_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="orders_2.html#E12:68_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:68_1_1_1_1_2_1_1_2"/>
<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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:68_1_1_1_1_2"/>
<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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E16:68_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/>


</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="E2:68_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="E3:68_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>7</sub></font>
 
;<br/>
<a NAME="E4:68_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>8</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <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="E5:68_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>2</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E21:68"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:68_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>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:68"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E19:68"><i><font color="Green">E54</font></i></a>, <a class="txt" href="orders_2.html#E20:68"><i><font color="Green">E55</font></i></a>, <a class="txt" href="orders_2.html#E21:68"><i><font color="Green">E56</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self">Def15</a></i>;<br/>


</div>
