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<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <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> ] means <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="E2:28_1_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> holds <font color="Olive">b<sub>1</sub></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="E3:28_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>4</sub></font>]
 
;<br/>
<a NAME="E4:28_1_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="grcat_1.html#K7">id</a> <font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></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="E6:28_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow21.html#V2" target="_self">lattice-wise</a>
 <b>and </b><br/><a NAME="E7:28_1_1"/><i><font color="Green">E14</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E8:28_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] ) )
 <b>from </b><i><a class="ref" href="yellow21.html#S1" target="_self">YELLOW21:sch 1</a>(<a class="txt" href="yellow21.html#E2:28_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="yellow21.html#E3:28_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yellow21.html#E4:28_1_1"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="altcat_1.html#V6">strict</a> <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a> <b>by </b><i><a class="txt" href="yellow21.html#E6:28_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D8" target="_self">YELLOW21:def 8</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>



<a NAME="E10:28_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V23">isomorphic</a> implies <font color="Maroon">c<sub>6</sub></font> is <a href="orders_3.html#V5">monotone</a> ) &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>
<b>hence </b><a NAME="E11:28_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V23">isomorphic</a> implies <font color="Maroon">c<sub>6</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow21.html#E7:28_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E8:28_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>


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