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

<b>let </b><font color="Maroon">A</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">f</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">A</font>;<br/><b>let </b><font color="Maroon">fC1</font> be    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">f</font>;<br/><b>let </b><font color="Maroon">fC2</font> be    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">f</font>;<br/>







<b>assume </b><a NAME="E1:61"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">fC1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">fC2</font>
 ;<br/>

<b>set </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> ) </span>}</span> ;<br/>
<a NAME="E2:61"/><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">fC1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:61"/><i><font color="Green">E39</font></i>: 
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">fC1</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E2:61_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font>
 <b>and </b><br/><a NAME="E3:61_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E4:61_2"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E5:61_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:61_2"/><b>then </b>
<font color="Maroon">a1</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">a2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font>
 ;<br/><a NAME="E7:61_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:61_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:61_2"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_2_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</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="61_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_2_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:61_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:61_2_1_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:61_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:61_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a>, </i>;<br/>
<b>hence </b><a NAME="E7:61_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, </i>;<br/>


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

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

<b>assume </b><a NAME="E2:61_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E4:61_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:61_2_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E6:61_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:61_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a>, </i>;<br/>
<b>hence </b><a NAME="E8:61_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:61_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:61"/><i><font color="Green">E80</font></i>: 
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">fC2</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E7:61"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fC2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:61_3"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E2:61_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font>
 <b>and </b><br/><a NAME="E3:61_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E4:61_4"/><i><font color="Green">E99</font></i>: 
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font> &amp;  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:61_4"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T36" target="_self">Th36</a>, , </i>;<br/><a NAME="E6:61_4"/><b>then </b>
<font color="Maroon">a1</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">a2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC1</font>
 ;<br/><a NAME="E7:61_4"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:61_4"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T36" target="_self">Th36</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:61_4"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_4_1"/>
 <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</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="61_4_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_4_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:61_4_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:61_4_1_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:61_4_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:61_4_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, </i>;<br/>
<b>hence </b><a NAME="E7:61_4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, </i>;<br/>


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

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

<b>assume </b><a NAME="E2:61_4_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E4:61_4_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:61_4_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E6:61_4_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">a2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:61_4_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, </i>;<br/>
<b>hence </b><a NAME="E8:61_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:61_4"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_5_1"/>
( ( <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC2</font> ) or ( <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC1</font> ) or ( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC1</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC2</font> ) or ( <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> &amp; <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> ) )
 </span><b>by </b><i>, , , <a class="ref" href="orders_2.html#D11" target="_self">Def11</a>, , , <a class="ref" href="orders_2.html#T29" target="_self">Th29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1_1"><b>suppose </b></a><a NAME="E1:61_5_1_1"/>
( <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC2</font> )
 ;<br/><div class="add"><a NAME="E2:61_5_1_1"/><b>then </b>
( <font color="Maroon">fC2</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> &amp; <font color="Maroon">fC1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/><b>hence </b><a NAME="E3:61_5_1_1"/>
( <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> iff <font color="Maroon">fC2</font> is  not    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1_2"><b>suppose </b></a><a NAME="E1:61_5_1_2"/>
( <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC1</font> )
 ;<br/><div class="add"><a NAME="E2:61_5_1_2"/><b>then </b>
( <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> &amp; <font color="Maroon">fC2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/><b>hence </b><a NAME="E3:61_5_1_2"/>
( <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> iff <font color="Maroon">fC2</font> is  not    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1_3"><b>suppose </b></a><a NAME="E1:61_5_1_3"/>
( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC1</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fC2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_5_1_3"/>
( <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> iff <font color="Maroon">fC2</font> is  not    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1_4"><b>suppose </b></a><a NAME="E1:61_5_1_4"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> &amp; <font color="Maroon">N</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> )
 ;<br/><div class="add"><a NAME="E2:61_5_1_4"/>
<font color="Maroon">fC1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:61_5_1_4"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font>
 <b>and </b><br/><a NAME="E5:61_5_1_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC1</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a>, </i>;<br/><a NAME="E6:61_5_1_4"/>
<font color="Maroon">fC2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/><b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E8:61_5_1_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC2</font>
 <b>and </b><br/><a NAME="E9:61_5_1_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC2</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a>, </i>;<br/><i><font color="Green">E110</font></i>: <font color="Maroon">a</font> = 
<font color="Maroon">f</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">fC2</font>,<font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="orders_2.html#T36" target="_self">Th36</a>, , , <a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, <a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>
;<br/><a NAME="E11:61_5_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC2</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T36" target="_self">Th36</a>, <a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:61_5_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E13:61_5_1_4"/>
( <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> iff <font color="Maroon">fC2</font> is  not    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> )
 <b>by </b><i>, <a class="txt" href="orders_2.html#E7"><i><font color="Green">Lemma29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:61"/>
( <font color="Maroon">fC1</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC2</font> iff <font color="Maroon">fC2</font> is  not    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">fC1</font> )
 ;<br/>


</div>
