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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</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>, <font color="Maroon" title="c3">F</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>assume </b><b>that </b><br/><a NAME="E1:99"/><i><font color="Green" title="E45">A1</font></i>: 
for <font color="Olive" title="b1">Y</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>  st <font color="Olive" title="b1">Y</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>  holds <br/> <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E2:99"/><i><font color="Green" title="E46">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/> ex <font color="Olive" title="b2">Y</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> st <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b2">Y</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">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="Olive" title="b2">Y</font>,<font color="Maroon" title="c1">L</font> )
 <b>and </b><br/><a NAME="E3:99"/><i><font color="Green" title="E47">A3</font></i>: 
for <font color="Olive" title="b1">Y</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>  st <font color="Olive" title="b1">Y</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>  holds <br/> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" target="_self" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E4:99"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">Y1</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="E6:99"/><i><font color="Green" title="E49">A5</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">Y1</font>,<font color="Maroon" title="c1">L</font> &amp; <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="c6">Y1</font>,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E2:99"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:99"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">Y2</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="E9:99"/><i><font color="Green" title="E50">A6</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">Y2</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c5">y</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="c7">Y2</font>,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E2:99"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">z</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y2</font></span>)</span>,<font color="Maroon" title="c1">L</font>;<br/>

<a NAME="E10:99"/><i><font color="Green" title="E51">A7</font></i>: 
( ( <font color="Maroon" title="c6">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c6">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y2</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>by </b><i><a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<b>hence </b><a NAME="E11:99"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E3:99"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="waybel_0.html#E4:99"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="waybel_0.html#E6:99"><i><font color="Green" title="E49">A5</font></i></a></i>;<br/>

<a NAME="E12:99"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y2</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c6">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y2</font> &amp; <font color="Maroon" title="c7">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y2</font> )
 
<b>by </b><i><a class="txt" href="waybel_0.html#E1:99"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="waybel_0.html#E6:99"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="waybel_0.html#E10:99"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E13:99"/>
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> &amp; <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E6:99"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="waybel_0.html#E9:99"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/>


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