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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">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="E1:50"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">S</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>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c2">S</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c3">T</font>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">T</font> is    <a href="orders_2.html#M1" title="ORDERS_2:mode.1">Initial_Segm</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1"/>
( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c2">S</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="c3">T</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>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E1:50"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_2"/>
<font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E1:50"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="orders_2.html#E2:50"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_3"><b>suppose </b></a><a NAME="E1:50_1_1_3"/><i><font color="Green" title="E37">A4</font></i>: 
( <font color="Maroon" title="c2">S</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="c3">T</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>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">a1</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="E3:50_1_1_3"/><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">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="c2">S</font>,<font color="Maroon" title="c4">a1</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E3:50"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="orders_2.html#D15" target="_self" title="ORDERS_2:def.15">Def15</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">a2</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="E5:50_1_1_3"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c2">S</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="c3">T</font>,<font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E2:50"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="orders_2.html#E1:50_1_1_3"><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/><a NAME="E6:50_1_1_3"/>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K2" title="ORDERS_2:func.2">LowerCone</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K2" title="ORDERS_2:func.2">LowerCone</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E3:50_1_1_3"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="orders_2.html#E5:50_1_1_3"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/><a NAME="E7:50_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">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="c5">a2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">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="c4">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <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="E8:50_1_1_3"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> : for <font color="Olive" title="b2">b</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">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="c5">a2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">b</font> </span>}</span>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a3</font> where <font color="Olive" title="b1">a3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> : for <font color="Olive" title="b2">b</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">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="c4">a1</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">b</font> </span>}</span>  )
 ;<br/><a NAME="E9:50_1_1_3"/><b>then </b><i><font color="Green" title="E41">A8</font></i>: 
 ex <font color="Olive" title="b1">a</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a1</font> &amp; ( for <font color="Olive" title="b2">b</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">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="c5">a2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">b</font> ) )
 ;<br/><a NAME="E10:50_1_1_3"/><i><font color="Green" title="E42">A9</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="c5">a2</font> &amp; ( for <font color="Olive" title="b2">b</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">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="c4">a1</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">b</font> ) )
 <b>by </b><i><a class="txt" href="orders_2.html#E8:50_1_1_3"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/><a NAME="E11:50_1_1_3"/>
( <font color="Maroon" title="c4">a1</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="c4">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">a2</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="c5">a2</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="E12:50_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a1</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E9:50_1_1_3"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="orders_2.html#E10:50_1_1_3"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E13:50_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T28" target="_self" title="ORDERS_2:th.28">Th28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:50"/>
contradiction
 ;<br/>


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