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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="setfam_1.html#V2">with_non-empty_element</a>  <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:80"/><i><font color="Green">E52</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D11">SETFAM_1:def 11</a></i>;<br/>
<a NAME="E2:80"/><i><font color="Green">E53</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E3:80"/><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K11" target="_self">-CL-opp_category</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E4:80"/><i><font color="Green">E55</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K11" target="_self">-CL-opp_category</a> </span>)</span>
 ;<br/><a NAME="E2:80_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K11" target="_self">-CL-opp_category</a> </span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> <b>by </b><i><a class="txt" href="waybel34.html#E3:80"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E4:80_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span>
 ;<br/><a NAME="E5:80_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_3.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E2:80"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel34.html#E1:80_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel34.html#D13" target="_self">Def13</a></i>;<br/><b>hence </b><a NAME="E6:80_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:80"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel34.html#D5" target="_self">Def5</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:80"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a> )
 ;<br/>

<a NAME="E7:80"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E4:80"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel34.html#D5" target="_self">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> <b>by </b><i><a class="ref" href="waybel34.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E9:80"/>
 <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>hence </b><a NAME="E10:80"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K11" target="_self">-CL-opp_category</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel34.html#E6:80"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel34.html#D13" target="_self">Def13</a></i>;<br/>


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