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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>;<br/>



<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:101_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font></span>)</span> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<a NAME="E2:101_1_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="waybel23.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:101_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i/>;<br/>

<a NAME="E4:101_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:101_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:101_1_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T52">YELLOW_2:52</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:101_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:101_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E2:101_1_2"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<a NAME="E3:101_1_2"/>
<font color="Maroon">x</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:101_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel23.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">x</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel23.html#T5" target="_self">Th5</a></i>;<br/><b>consider </b><font color="Maroon">J</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:101_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E5:101_2"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</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">L</font> <b>by </b><i>, <a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/><a NAME="E7:101_2"/><i><font color="Green">E48</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">J</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">I</font>,<font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:101"/>
 <a href="waybel23.html#K3" target="_self">supMap</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>


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