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<div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>;<br/>
<a NAME="E1:53_1_1"/>
<span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span> is <a href="waybel_0.html#V12">lower</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a><br/>

<b>assume </b><a NAME="E1:53_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">S1</font>, <font color="Maroon">s2</font> = <font color="Maroon">S2</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s1</font>;<br/>
<b>set </b><font color="Maroon">C2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s2</font>;<br/>
<a NAME="E5:53_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:53_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y1</font> = <font color="Maroon">y1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">s1</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font>, <font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">s2</font> <b>by </b><i>, </i>;<br/>
<a NAME="E10:53_1_1_1"/>
( <font color="Maroon">x1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x2</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E11:53_1_1_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>
<b>hence </b><a NAME="E12:53_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_1_1"/>
<span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span> is  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 ;<br/>


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