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<div class="add">

<a NAME="E1:26_1"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">F</font> being <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>( ( for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <font color="Olive">F</font> <a href="functor0.html#K8">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E2:26_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:26_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> )
 
;<br/>
<b>hence </b><a NAME="E2:26_1_1"/>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font>  st <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">g</font>) holds <br/><font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:26_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>, <font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E2:26_1_2"/>
(  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:26_1_2"/>
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">g</font>) implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>assume </b><a NAME="E1:26_1_3"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:26_1_3"/><i><font color="Green">E54</font></i>: 
(  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:26_1_3"/>
<font color="Maroon">P<sub>2</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font>, <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font>, <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font>]
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">d</font> being   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font><b> such that </b><br/><a NAME="E5:26_1_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>] &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c</font>) &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font>, <font color="Maroon">d'</font> = <font color="Maroon">d</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/><a NAME="E7:26_1_3"/>
<font color="Maroon">g</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'</font>,<font color="Maroon">d'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:26_1_3"/>
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">c</font>,<font color="Olive">d</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">c</font>) &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">g</font>) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:26_1"/>
<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <a href="functor0.html#R2">are_anti-isomorphic</a> 
 <b>from </b><i><a class="ref" href="yellow18.html#S13">YELLOW18:sch 13</a>(, , , );</i><br/>


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