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









<b>assume </b><b>that </b><br/><a NAME="E1:51"/><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>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:51"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E4:51"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">T</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">S</font>
 ;<br/>

<b>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="E6:51"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E7:51"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">S</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T29" target="_self">Th29</a>, <a class="ref" href="orders_2.html#T32" target="_self">Th32</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:51_1"/>
<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>
 ;<br/><a NAME="E2:51_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:51_1"/>
<font color="Maroon">a2</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">a</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#T30" target="_self">Th30</a>, <a class="txt" href="orders_2.html#E3:51"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E4:51_1"/><b>then </b>
<font color="Maroon">a2</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">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:51_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a3</font> where <font color="Olive">a3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">c</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> holds <br/><font color="Olive">a3</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">c</font> </span>}</span> 
 ;<br/><a NAME="E6:51_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">a3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">c</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> holds <br/><font color="Olive">a3</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">c</font> ) )
 ;<br/><a NAME="E7:51_1"/>
<font color="Maroon">a</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/><a NAME="E8:51_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E7:51"><i><font color="Green">E41</font></i></a></i>;<br/><b>hence </b><a NAME="E9:51_1"/>
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T28" target="_self">Th28</a>, <a class="txt" href="orders_2.html#E4:51"><i><font color="Green">E40</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E9:51"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">c</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> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">c</font> </span>}</span> 
 
;<br/>
<a NAME="E10:51"/><b>then </b>
<font color="Maroon">a1</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">a</font></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
<a NAME="E11:51"/><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">a</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#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:51"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E3:51"><i><font color="Green">E39</font></i></a></i>;<br/>


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