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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="E1:50"/><i><font color="Green">E20</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/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:50"/>
<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="yellow21.html#K6" target="_self">-UPS_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="yellow21.html#K6" target="_self">-UPS_category</a> </span>)</span> ;<br/>
<a NAME="E4:50"/>
 <a href="yellow21.html#K4" target="_self">latt</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E5:50"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4" target="_self">latt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:50"><i><font color="Green">E20</font></i></a>, <a class="ref" href="yellow21.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:50"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow21.html#K2" target="_self">POSETS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow21.html#D2" target="_self">Def2</a></i>;<br/>


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