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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="E1:53"/><i><font color="Green">E23</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>, <font color="Maroon">c<sub>3</sub></font> be   <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/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ;<br/><a NAME="E3:53_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:53_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow21.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon">c<sub>4</sub></font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4" target="_self">latt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<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>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:53"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yellow21.html#E1:53_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow21.html#D10" target="_self">Def10</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:53"/>
( <font color="Maroon">c<sub>4</sub></font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4" target="_self">latt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<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> implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow21.html#E1:53"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow21.html#D10" target="_self">Def10</a></i>;<br/>


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