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<div class="add">

<b>let </b><font color="Maroon">DP</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_3.html#V1" target="_self">discrete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DP</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DP</font><b> such that </b><a NAME="E1:15"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:15"/><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">DP</font> ;<br/>
<a NAME="E5:15"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">DP</font> <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green">E31</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="wellord1.html#D6">WELLORD1:def 6</a></i>;<br/>
<a NAME="E8:15"/><i><font color="Green">E40</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="wellord1.html#D6">WELLORD1:def 6</a></i>;<br/>
<a NAME="E9:15"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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="E1:15_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E5:15_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:15_1_1_1"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, , <a class="txt" href="orders_3.html#E2"><i><font color="Green">Lemma17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font>
 <b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E2"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:15_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>;<br/><a NAME="E5:15_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E6:15_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:15_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E8:15_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:15"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E11:15"/><b>then </b><i><font color="Green">E50</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">DP</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E12:15"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">DP</font> ;<br/>
<a NAME="E14:15"/>
<font color="Maroon">C</font> is <a href="orders_3.html#V2" target="_self">disconnected</a>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E15:15"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">DP</font> is <a href="orders_3.html#V2" target="_self">disconnected</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>hence </b><a NAME="E16:15"/>
<font color="Maroon">DP</font> is <a href="orders_3.html#V3" target="_self">disconnected</a>
 <b>by </b><i/>;<br/>


</div>
