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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <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> holds   <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="Olive" title="b1">f</font></span>)</span> is    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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="c2">f</font></span>)</span> is    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c2">f</font></span><br/>



<b>reconsider </b><font color="Maroon" title="c3">C</font> =  <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="c2">f</font></span>)</span> as   <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="orders_2.html#E104"><i><font color="Green" title="E44">Lm3</font></i></a></i>;<br/>
<a NAME="E2:75"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c3">C</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="ref" href="orders_2.html#T87" target="_self" title="ORDERS_2:th.87">Th87</a></i>;<br/>
<a NAME="E3:75"/><i><font color="Green" title="E49">A2</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <font color="Maroon" title="c3">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<a NAME="E1:75_1"/>
( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self" title="ORDERS_2:def.4">Def4</a>, <a class="ref" href="orders_2.html#D5" target="_self" title="ORDERS_2:def.5">Def5</a>, <a class="ref" href="orders_2.html#D6" target="_self" title="ORDERS_2:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E2:75_1"/>
( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c3">C</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c3">C</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c3">C</font> )
 <b>by </b><i><a class="ref" href="orders_1.html#T93" title="ORDERS_1:th.93">ORDERS_1:93</a>, <a class="ref" href="orders_1.html#T94" title="ORDERS_1:th.94">ORDERS_1:94</a>, <a class="ref" href="orders_1.html#T95" title="ORDERS_1:th.95">ORDERS_1:95</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c3">C</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<a NAME="E3:75_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self" title="ORDERS_2:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92" title="ORDERS_1:th.92">ORDERS_1:92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> <font color="Maroon" title="c3">C</font></span><br/>

<b>let </b><font color="Maroon" title="c4">Y</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="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> or <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> ) )</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="c4">Y</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E6:75_1"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E7:75_1"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c4">Y</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<a NAME="E8:75_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E6:75_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="orders_2.html#E7:75_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="E10:75_1"/><i><font color="Green" title="E52">A5</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="E11:75_1"/><i><font color="Green" title="E53">A6</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="c2">f</font>
 <b>by </b><i><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="c2">f</font> <b>by </b><i><a class="txt" href="orders_2.html#E11:75_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="orders_2.html#D17" target="_self" title="ORDERS_2:def.17">Def17</a></i>;<br/>
<a NAME="E13:75_1"/>
( <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">Y</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>  &amp; <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:75_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="orders_2.html#E10:75_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="orders_2.html#D16" target="_self" title="ORDERS_2:def.16">Def16</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:75_1"/><i><font color="Green" title="E54">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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">Y</font>
 <b>and </b><br/><a NAME="E16:75_1"/><i><font color="Green" title="E55">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T9" title="WELLORD1:th.9">WELLORD1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">aa</font> = <font color="Maroon" title="c8">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="orders_2.html#E15:75_1"><i><font color="Green" title="E54">A7</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<b>thus </b><a NAME="E18:75_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E15:75_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font></span><br/>

<b>thus </b><a NAME="E19:75_1"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E1:75_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:75_1_1"/><b>then </b><i><font color="Green" title="E56">A9</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</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/>
<a NAME="E3:75_1_1"/><b>then </b>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E6:75_1"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E5:75_1_1"/><i><font color="Green" title="E57">A10</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c8">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:75_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:75_1_1"/><b>then </b><i><font color="Green" title="E58">A11</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">aa</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self" title="ORDERS_2:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_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="E1:75_1_1_1_1"/>
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">C</font> or <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_1"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font></span><br/><div class="add"><a NAME="E2:75_1_1_1_1_1"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c11">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c9">aa</font> &amp; <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="c3">C</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E6:75_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="orders_2.html#E15:75_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="orders_2.html#E2:75_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="orders_2.html#E5:75_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="orders_2.html#E6:75_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a>, <a class="ref" href="orders_2.html#T88" target="_self" title="ORDERS_2:th.88">Th88</a>, <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="E3:75_1_1_1_1_1"/>
<font color="Maroon" title="c10">y</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="orders_2.html#T70" target="_self" title="ORDERS_2:th.70">Th70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_2"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:75_1_1_1_1_2"/>
<font color="Maroon" title="c10">y</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#E6:75_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="orders_2.html#E2:75_1_1"><i><font color="Green" title="E56">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:75_1_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E2:75_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:75_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E16:75_1"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/>
<a NAME="E10:75_1_1"/><b>then </b>
<font color="Maroon" title="c9">aa</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">b</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self" title="ORDERS_2:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E11:75_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="orders_2.html#E5:75_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="orders_2.html#E6:75_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="orders_2.html#T25" target="_self" title="ORDERS_2:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E20:75_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> implies <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/><b>assume </b><a NAME="E1:75_2"/><i><font color="Green" title="E50">A12</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/><b>then consider </b><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:75_2"/><i><font color="Green" title="E51">A13</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E4:75_2"/><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon" title="c5">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="c2">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> as    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c2">f</font> <b>by </b><i><a class="txt" href="orders_2.html#E4:75_2"><i><font color="Green" title="E52">A14</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="75_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_2_1_1"/>
( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> or <font color="Maroon" title="c6">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1_1_1"><b>suppose </b></a><a NAME="E1:75_2_1_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/><div class="add"><b>hence </b><a NAME="E2:75_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:75_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="orders_2.html#D16" target="_self" title="ORDERS_2:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1_1_2"><b>suppose </b></a><a NAME="E1:75_2_1_1_2"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/><div class="add"><a NAME="E2:75_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">X</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T88" target="_self" title="ORDERS_2:th.88">Th88</a></i>;<br/><a NAME="E3:75_2_1_1_2"/><b>then </b>
(  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">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="c6">X</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E3:75_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="orders_2.html#D16" target="_self" title="ORDERS_2:def.16">Def16</a>, <a class="ref" href="orders_2.html#T71" target="_self" title="ORDERS_2:th.71">Th71</a></i>;<br/><b>hence </b><a NAME="E4:75_2_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:75_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" title="ORDERS_2:func.1">UpperCone</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:75"/>
 <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="c2">f</font></span>)</span> is    <a href="orders_2.html#M2" title="ORDERS_2:mode.2">Chain</a> of <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:75"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="orders_2.html#E3:75"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="orders_2.html#D16" target="_self" title="ORDERS_2:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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