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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="compts_1.html#NV6" title="COMPTS_1:NV.6">Hausdorff</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E50">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Olive" title="b1">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<b>thus </b><a NAME="E2:98"/>
<font color="Maroon" title="c1">S</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D6" title="YELLOW_0:def.6">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" title="c2">x</font> =  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c2">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D4" title="YELLOW_0:def.4">YELLOW_0:def 4</a><br/>

<i><font color="Green" title="E51">A2</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self" title="WAYBEL_9:def.6">Def6</a></i>
<br/>.= 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D11" title="GRCAT_1:def.11">GRCAT_1:def 11</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>
;<br/>
<a NAME="E3:98_1"/>
 <a href="waybel_9.html#R2" title="WAYBEL_9:pred.2">ex_inf_of</a> <font color="Maroon" title="c1">S</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:98"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="waybel_9.html#T39" target="_self" title="WAYBEL_9:th.39">Th39</a></i>;<br/>
<a NAME="E4:98_1"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:98_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="waybel_9.html#D4" target="_self" title="WAYBEL_9:def.4">Def4</a></i>;<br/>
<a NAME="E5:98_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:98_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:98_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E4:98_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>


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

<b>reconsider </b><font color="Maroon" title="c2">x</font> =  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+id</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c2">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5" title="YELLOW_0:def.5">YELLOW_0:def 5</a><br/>

<i><font color="Green" title="E51">A4</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+id</a> </span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_9.html#D5" target="_self" title="WAYBEL_9:def.5">Def5</a></i>
<br/>.= 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D11" title="GRCAT_1:def.11">GRCAT_1:def 11</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>
;<br/>
<a NAME="E5:98"/>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Maroon" title="c1">S</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+id</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:98"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="waybel_9.html#T38" target="_self" title="WAYBEL_9:th.38">Th38</a></i>;<br/>
<a NAME="E6:98"/><b>then </b><i><font color="Green" title="E52">A5</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:98"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="waybel_9.html#D3" target="_self" title="WAYBEL_9:def.3">Def3</a></i>;<br/>
<a NAME="E7:98"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:98"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:98"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E6:98"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>


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