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<b>consider </b><font color="Maroon">L</font> being  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E2:10_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> holds  <font color="Olive">a</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:10_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="waybel_0.html#NM8">LATTICE</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">a</font>,<font color="Olive">b</font><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">b</font>,<font color="Olive">c</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">g</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>,<font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font>]
 
;<br/>
<a NAME="E4:10_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font> being  <a href="waybel_0.html#NM8">LATTICE</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">a</font>, <a href="grcat_1.html#K7">id</a> <font color="Olive">a</font>]
 
;<br/>
<b>consider </b><font color="Maroon">C</font> being  <a href="altcat_1.html#V6">strict</a> <a href="altcat_1.html#NM4">category</a><b> such that </b><br/><a NAME="E6:10_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">C</font> is <a href="yellow21.html#V2" target="_self">lattice-wise</a>
 <b>and </b><br/><a NAME="E7:10_1_1"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E8:10_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="waybel_0.html#NM8">LATTICE</a><br/> for <font color="Olive">f</font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] ) )
 <b>from </b><i><a class="ref" href="yellow21.html#S1" target="_self">YELLOW21:sch 1</a>(, , );<br/></i>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E9:10_1_1"/>
<font color="Maroon">C</font> is <a href="altcat_1.html#V6">strict</a>
 ;<br/>

<b>thus </b><a NAME="E10:10_1_1"/>
<font color="Maroon">C</font> is <a href="yellow21.html#V2" target="_self">lattice-wise</a>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D5" target="_self">YELLOW21:def 5</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>

<b>thus </b><a NAME="E11:10_1_1"/>
<font color="Maroon">a</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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