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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c4">fC</font> be    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c3">f</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c4">fC</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">fC</font>;<br/>
<a NAME="E4:74"/>
<font color="Maroon" title="c4">fC</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D17" target="_self" title="ORDERS_2:def.17">Def17</a></i>;<br/>
<a NAME="E5:74"/><b>then </b>
<font color="Maroon" title="c4">fC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E6:74"/><b>then </b>
not  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fC</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:74"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:74"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">fC</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:74"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E8:74"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:74"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>and </b><br/><a NAME="E11:74"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:74"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c3">f</font> <b>by </b><i><a class="txt" href="orders_2.html#E11:74"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="orders_2.html#D17" target="_self" title="ORDERS_2:def.17">Def17</a></i>;<br/>
<a NAME="E13:74"/><i><font color="Green" title="E52">A6</font></i>: 
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fC</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:74"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:74"/>
( <font color="Maroon" title="c4">fC</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">X</font> &amp; not <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fC</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:74"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="orders_2.html#E10:74"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E15:74"/><b>then </b>
not <font color="Maroon" title="c7">X</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">fC</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E16:74"/><b>then </b>
<font color="Maroon" title="c7">X</font> is  not    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c4">fC</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T84" target="_self" title="ORDERS_2:th.84">Th84</a></i>;<br/>
<a NAME="E17:74"/><b>then </b>
( <font color="Maroon" title="c4">fC</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="orders_2.html#E10:74"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="orders_2.html#E13:74"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="orders_2.html#T83" target="_self" title="ORDERS_2:th.83">Th83</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E19:74"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>and </b><br/><a NAME="E20:74"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c4">fC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/>
<a NAME="E21:74"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:74"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="orders_2.html#E11:74"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E22:74"/>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:74_1"/>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green" title="E56">A10</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">a</font>
 ;<br/>

<a NAME="E2:74_1_1"/><b>then </b><i><font color="Green" title="E57">A11</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K2" title="ORDERS_2:func.2">LowerCone</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:74_1_1"/><i><font color="Green" title="E58">A12</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>and </b><br/><a NAME="E5:74_1_1"/><i><font color="Green" title="E59">A13</font></i>: 
for <font color="Olive" title="b1">a2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Maroon" title="c10">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">a2</font>
 ;<br/>
<a NAME="E6:74_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:74_1_1"/><b>then </b><i><font color="Green" title="E60">A14</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E5:74_1_1"><i><font color="Green" title="E59">A13</font></i></a></i>;<br/>
<a NAME="E8:74_1_1"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:74_1_1"><i><font color="Green" title="E56">A10</font></i></a>, <a class="txt" href="orders_2.html#E4:74_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:74_1_1"/><i><font color="Green" title="E61">A15</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font>
 <b>and </b><br/><a NAME="E11:74_1_1"/><i><font color="Green" title="E62">A16</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K4" title="ORDERS_2:func.4">Chains</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:74"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c11">Y</font> as    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c3">f</font> <b>by </b><i><a class="txt" href="orders_2.html#E11:74_1_1"><i><font color="Green" title="E62">A16</font></i></a>, <a class="ref" href="orders_2.html#D17" target="_self" title="ORDERS_2:def.17">Def17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_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="E1:74_1_1_1_1"/>
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Y</font> or <font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">Y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_1"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:74_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="orders_2.html#E4:74_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="orders_2.html#E10:74_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_2"/><i><font color="Green" title="E63">A17</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">Y</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_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="E1:74_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c7">X</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c12">Y</font> or <font color="Maroon" title="c12">Y</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c7">X</font> )
 </span><b>by </b><i><a class="txt" href="orders_2.html#E1:74_1_1_1_1_2"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="orders_2.html#T83" target="_self" title="ORDERS_2:th.83">Th83</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">X</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c12">Y</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E19:74"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="orders_2.html#E4:74_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="orders_2.html#E7:74_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="orders_2.html#E10:74_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="orders_2.html#T70" target="_self" title="ORDERS_2:th.70">Th70</a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:74_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">Y</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c7">X</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">Y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T84" target="_self" title="ORDERS_2:th.84">Th84</a></i>;<br/><a NAME="E3:74_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:74_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:74_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="orders_2.html#E4:74_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="orders_2.html#E10:74_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:74_1_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:74_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:74_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">a</font>
 ;<br/>

<a NAME="E3:74_1"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K2" title="ORDERS_2:func.2">LowerCone</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E21:74"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:74"/>
<font color="Maroon" title="c4">fC</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E19:74"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="orders_2.html#E20:74"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="orders_2.html#E21:74"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/>


</div>
