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<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">L2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E34</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L2</font> #) &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>  st  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a><br/>

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a><br/>

<b>assume </b><a NAME="E3:22"/><i><font color="Green">E43</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">E</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<a NAME="E5:22"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E8:22"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E9:22"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y2</font>
;<br/>

<b>thus </b><a NAME="E11:22"/>
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E13:22"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">x2</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y2</font> )
 ;<br/>

<a NAME="E14:22"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E15:22"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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