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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>thus </b><a NAME="E2:26"/>
 <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c3">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> )
 ;<br/>
<a NAME="E4:26_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">Z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E3:26_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="yellow_7.html#T11" target="_self" title="YELLOW_7:th.11">Th11</a>, <a class="ref" href="yellow_7.html#T13" target="_self" title="YELLOW_7:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E5:26_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:26"/>
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c2">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">Y</font><b> such that </b><br/><a NAME="E3:26_2"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">Z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
 ;<br/>
<a NAME="E4:26_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E3:26_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="yellow_7.html#T11" target="_self" title="YELLOW_7:th.11">Th11</a>, <a class="ref" href="yellow_7.html#T13" target="_self" title="YELLOW_7:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E5:26_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:26"/>
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c3">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:26_3"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> )
 ;<br/>
<a NAME="E4:26_3"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c5">Z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E3:26_3"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="yellow_7.html#T10" target="_self" title="YELLOW_7:th.10">Th10</a>, <a class="ref" href="yellow_7.html#T12" target="_self" title="YELLOW_7:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E5:26_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:26"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">Y</font><b> such that </b><br/><a NAME="E7:26"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c5">Z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
 ;<br/>
<a NAME="E8:26"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E7:26"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="yellow_7.html#T10" target="_self" title="YELLOW_7:th.10">Th10</a>, <a class="ref" href="yellow_7.html#T12" target="_self" title="YELLOW_7:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E9:26"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


</div>
