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<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">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>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:46"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/>

<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="E2:46"/>
<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">S</font>,<font color="Maroon">a1</font>
 ;<br/>

<a NAME="E3:46"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</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">a1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 
;<br/>
<a NAME="E4:46"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self">Def10</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:46"/><b>then </b>
<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> : for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">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">a1</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b</font> </span>}</span> 
 
;<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="E8:46"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E9:46"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">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">a1</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b</font>
 ;<br/>
<a NAME="E10:46"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:46"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E12:46"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self">Def9</a>, </i>;<br/>
<a NAME="E13:46"/><b>then </b>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">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">a2</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:46"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">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">a2</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">c</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b</font> </span>}</span> 
 
;<br/>
<a NAME="E15:46"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E16:46"/><b>then </b>
<font color="Maroon">x</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">S</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:46"/>
<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">S</font>,<font color="Maroon">a2</font>
 ;<br/>


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