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

<a NAME="E1:24"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">x</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</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">a</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">b</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>



<b>set </b><font color="Maroon">A</font> = <font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">f</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">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>, <font color="Maroon">o2</font> being   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span><b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</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">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">f</font> is   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:24_1"/>
( <font color="Maroon">a</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">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> &amp; <font color="Maroon">b</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">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:24_1"/>
( <font color="Maroon">a</font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">b</font> is <a href="lattice3.html#V3">complete</a> )
 <b>by </b><i>, , </i>;<br/><a NAME="E7:24_1"/>
(  <a href="yellow21.html#K3">latt</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3">latt</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><b>hence </b><a NAME="E8:24_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:24"/>
( <font color="Maroon">a</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">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> &amp; <font color="Maroon">b</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">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> ;<br/>
<a NAME="E5:24"/><i><font color="Green">E46</font></i>: 
(  <a href="yellow21.html#K3">latt</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3">latt</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>assume </b><a NAME="E6:24"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">b</font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 ;<br/>

<a NAME="E7:24"/><b>then </b>
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T17">WAYBEL_1:17</a></i>;<br/>
<a NAME="E8:24"/><b>then </b>
<font color="Maroon">f</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">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E9:24"/>
<font color="Maroon">f</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">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>;<br/>


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