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<b>let </b><font color="Maroon" title="c1">L</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c2">A</font> holds <br/><font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">X</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E4:59"/>
<font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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#D20" target="_self" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a><br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel_0.html#E3:59_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<a NAME="E5:59_1"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c8">Y</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E1:59"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="waybel_0.html#E3:59_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:59_1"/>
<font color="Maroon" title="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<a NAME="E7:59_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E3:59_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="waybel_0.html#E5:59_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="waybel_0.html#D20" target="_self" title="WAYBEL_0:def.20">Def20</a></i>;<br/>
<b>hence </b><a NAME="E8:59_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E3:59_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>


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