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<b>let </b><font color="Maroon">L1</font> be    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L2</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green">E28</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> #)
 ;<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><a NAME="E2:39"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>thus </b><a NAME="E3:39"/>
 <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/>
<b>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="E5:39_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y'</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/>
<a NAME="E7:39_1"/>
<font color="Maroon">y'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E4:39"/>
 <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E5:39_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<a NAME="E7:39_2"/>
<font color="Maroon">y'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E5:39"/>
 <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/>
<b>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="E5:39_3"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y'</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/>
<a NAME="E7:39_3"/>
<font color="Maroon">y'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:39_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E6:39"/>
 <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E5:39_4"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<a NAME="E7:39_4"/>
<font color="Maroon">y'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:39_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


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


</div>
