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<b>let </b><font color="Maroon">S</font> be  <a href="compts_1.html#V2">compact</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">x</font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>thus </b><a NAME="E2:98"/>
<font color="Maroon">S</font> is <a href="yellow_0.html#V1">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D6">YELLOW_0:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">x</font> =  <a href="waybel_9.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>take </b>
<font color="Maroon">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D4">YELLOW_0:def 4</a><br/>

<i><font color="Green">E34</font></i>:  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
;<br/>
<a NAME="E3:98_1"/>
 <a href="waybel_9.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">S</font> <a href="waybel_9.html#K4" target="_self">opp+id</a> 
 
<b>by </b><i>, <a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<a NAME="E4:98_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:98_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:98_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>


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

<b>reconsider </b><font color="Maroon">x</font> =  <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="waybel_9.html#K3" target="_self">+id</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>take </b>
<font color="Maroon">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a><br/>

<i><font color="Green">E36</font></i>:  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="waybel_9.html#K3" target="_self">+id</a> </span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
;<br/>
<a NAME="E5:98"/>
 <a href="waybel_9.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">S</font> <a href="waybel_9.html#K3" target="_self">+id</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:98"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:98"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:98"/>
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


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