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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="E1:26"/><i><font color="Green">E17</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="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <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> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> is   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="waybel34.html#T12" target="_self">Th12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="waybel34.html#E3:26_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>thus </b><a NAME="E5:26_1"/>
( <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#K5" target="_self">-SUP_category</a> </span>)</span> &amp; <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#K5" target="_self">-SUP_category</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel34.html#E3:26_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>thus </b><a NAME="E6:26_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="lattice3.html#V3">complete</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:26"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel34.html#E3:26_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="waybel34.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E7:26_1"/>
(  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="yellow21.html#K5">@</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E3:26_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><b>hence </b><a NAME="E8:26_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:26"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel34.html#E3:26_1"><i><font color="Green">E18</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="E3:26"/>
( <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#K5" target="_self">-SUP_category</a> </span>)</span> &amp; <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#K5" target="_self">-SUP_category</a> </span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</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#K5" target="_self">-SUP_category</a> </span>)</span> ;<br/>
<a NAME="E5:26"/><i><font color="Green">E18</font></i>: 
(  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>assume </b><a NAME="E6:26"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> )
 ;<br/>

<a NAME="E7:26"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T18">WAYBEL_1:18</a></i>;<br/>
<a NAME="E8:26"/><b>then </b>
<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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:26"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel34.html#E5:26"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel34.html#E6:26"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel34.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E9:26"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>;<br/>


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