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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:38"/><i><font color="Green">E28</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E3:38"/><i><font color="Green">E29</font></i>: 
 <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel31.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1"/>
(  not <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/><div class="add"><a NAME="E2:38_1_1_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E2:38"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E3:38_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><b>hence </b><a NAME="E4:38_1_1_1"/>
 <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="waybel31.html#K1" target="_self">CLweight</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel31.html#E4:38"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel31.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/><div class="add"><a NAME="E2:38_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E2:38"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E3:38_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E4:38_1_1_2"/><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel31.html#E4:38"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel31.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E5:38_1_1_2"/>
 <a href="waybel23.html#K2">weight</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:38_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel31.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E6:38_1_1_2"/>
 <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="waybel31.html#K1" target="_self">CLweight</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel31.html#E4:38"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel31.html#E3:38_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel31.html#E4:38_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel31.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:38"/>
 <a href="waybel31.html#K1" target="_self">CLweight</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="waybel31.html#K1" target="_self">CLweight</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>


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