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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">a1</font>, <font color="Maroon" title="c3">a2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">T</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>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c5">T</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c4">S</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">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="E6:57"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <b>and </b><br/><a NAME="E7:57"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c5">T</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="c4">S</font>,<font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:57"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="orders_2.html#E4:57"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c7">b</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="c6">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E2:57_1"/><b>then </b><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:57_1"/>
<font color="Maroon" title="c3">a2</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="c6">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="orders_2.html#E3:57"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="orders_2.html#E7:57"><i><font color="Green" title="E38">A5</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="E4:57_1"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
 ex <font color="Olive" title="b1">a3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> &amp; ( for <font color="Olive" title="b2">c</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="b2">c</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="c6">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">a3</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">c</font> ) )
 ;<br/><a NAME="E5:57_1"/>
<font color="Maroon" title="c6">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="c6">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="E6:57_1"/><b>then </b>
<font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E4:57_1"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E7:57_1"/>
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:57"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="orders_2.html#E2:57_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="orders_2.html#T29" target="_self" title="ORDERS_2:th.29">Th29</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:57"/><b>then </b>
<font color="Maroon" title="c2">a1</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="c6">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
;<br/>
<b>hence </b><a NAME="E10:57"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:57"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="orders_2.html#E7:57"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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