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<div class="add">

<b>thus </b><a NAME="E1:57_1"/>
( <font color="Maroon">F<sub>4</sub></font>() is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>5</sub></font>() is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">F<sub>2</sub></font>() )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self">YELLOW20:def 5</a><br/>

<a NAME="E2:57_1"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>()) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>(),  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>(),  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><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>4</sub></font>();<br/><a NAME="E1:57_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>4</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:57_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:57_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>) is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>()
 <b>by </b><i><a class="txt" href="yellow20.html#E3:57"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><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>4</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><b>assume </b><a NAME="E1:57_1_2"/><i><font color="Green">E52</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/><a NAME="E2:57_1_2"/>
( <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>4</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>4</sub></font>() )
 ;<br/><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">a</font>, <font color="Maroon">d</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a></i>;<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="E5:57_1_2"/><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="tarski.html#R1">c=</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> &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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then 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">c</font>,<font color="Maroon">d</font> ;<br/><a NAME="E7:57_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, , <a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E8:57_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K8">.</a> <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">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:57_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E9:57_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>5</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>)
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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>4</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>(), <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><b>assume </b><a NAME="E1:57_1_3"/><i><font color="Green">E55</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</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/><a NAME="E2:57_1_3"/>
( <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>4</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>4</sub></font>() &amp; <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>4</sub></font>() )
 ;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>;<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/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>(), <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>(), <font color="Maroon">c'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>();<br/><b>assume </b><a NAME="E4:57_1_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>) &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c</font>) )
 ;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">a'</font>;<br/><b>let </b><font color="Maroon">g'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c'</font>,<font color="Maroon">b'</font>;<br/><b>assume </b><a NAME="E5:57_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>) )
 ;<br/><a NAME="E6:57_1_3"/><i><font color="Green">E58</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="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">b1</font> ;<br/><a NAME="E8:57_1_3"/><i><font color="Green">E59</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">c1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">c1</font> ;<br/><a NAME="E10:57_1_3"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</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> 
 <b>by </b><i><a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/><a NAME="E11:57_1_3"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>;<br/><a NAME="E12:57_1_3"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font> )
 <b>by </b><i><a class="ref" href="yellow20.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E13:57_1_3"/><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a>, , , , <a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E14:57_1_3"/><b>then </b><i><font color="Green">E134</font></i>: 
( <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">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E2:57_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, <a class="ref" href="yellow20.html#T11" target="_self">Th11</a>, </i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font>) = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, <a class="ref" href="yellow20.html#T11" target="_self">Th11</a>, , <a class="ref" href="functor0.html#D25">FUNCTOR0:def 25</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g'</font>
<b>by </b><i>, , , <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E135</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_4"><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>4</sub></font>();<br/><b>let </b><font color="Maroon">a'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>();<br/><b>assume </b><a NAME="E1:57_1_4"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>)
 ;<br/><a NAME="E2:57_1_4"/>
<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>4</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:57_1_4"/>
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</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">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a1</font> )
 <b>by </b><i><a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>;<br/><b>hence </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">a</font>, <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font>) = 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="altcat_4.html#T13">ALTCAT_4:13</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="yellow20.html#D1" target="_self">Def1</a>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
;<br/><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()<b> such that </b><br/><a NAME="E8:57_1"/><i><font color="Green">E137</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>() holds  <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 <b>and </b><br/><a NAME="E9:57_1"/><i><font color="Green">E138</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>4</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="Maroon">G</font> <a href="functor0.html#K8">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</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="yellow18.html#S9">YELLOW18:sch 9</a>(<a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, , <a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="txt" href="yellow20.html#E3:57_1"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E10:57_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V19">injective</a> &amp; <font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V20">surjective</a> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E2:57"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E11:57_1"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V4">one-to-one</a> &amp; <font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V17">faithful</a> &amp; <font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V20">surjective</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#D34">FUNCTOR0:def 34</a></i>;<br/>
<div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_5"><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>4</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><a NAME="E1:57_1_5"/>
( <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>4</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>4</sub></font>() )
 ;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>;<br/><b>assume </b><a NAME="E3:57_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>)
 ;<br/><b>then </b><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E5:57_1_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow20.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_6"><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>4</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><b>assume </b><a NAME="E1:57_1_6"/><i><font color="Green">E143</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/><a NAME="E2:57_1_6"/>
( <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>4</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>4</sub></font>() )
 ;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>;<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="E4:57_1_6"/><i><font color="Green">E144</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="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">b1</font> ;<br/><b>assume </b><a NAME="E6:57_1_6"/>
<font color="Maroon">H<sub>2</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">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">g</font>)
 ;<br/><b>then </b><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g1</font>
<b>by </b><i>, <a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:57_1_6"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="yellow20.html#T11" target="_self">Th11</a>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E14:57_1"/><i><font color="Green">E145</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>5</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>  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>4</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="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>) &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</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="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">g</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_7"><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>5</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>5</sub></font>();<br/>

<b>assume </b><a NAME="E1:57_1_7"/><i><font color="Green">E146</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/>

<a NAME="E2:57_1_7"/>
( <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>5</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>5</sub></font>() &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>5</sub></font>() <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() ;<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="E4:57_1_7"/><i><font color="Green">E147</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="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">b1</font> ;<br/>
<b>consider </b><font color="Maroon">c1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">d1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">g1</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c1</font>,<font color="Maroon">d1</font><b> such that </b><br/><a NAME="E7:57_1_7"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">d1</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">d1</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">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c1</font>, <font color="Maroon">d</font> = <font color="Maroon">d1</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E3:57"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E9:57_1_7"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">g1</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><a class="txt" href="yellow20.html#E2:57_1"><i><font color="Green">E34</font></i></a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font> <b>by </b><i><a class="txt" href="yellow20.html#E2:57_1"><i><font color="Green">E34</font></i></a>, , </i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E11:57_1_7"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c</font>) &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">d</font>) &amp; <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> <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">H<sub>2</sub></font>(<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>) )
 <b>by </b><i>, , , <a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E15:57_1"/>
<font color="Maroon">G</font> is <a href="functor0.html#V21">bijective</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S12">YELLOW18:sch 12</a>(<a class="ref" href="yellow20.html#D1" target="_self">Def1</a>, , , , );</i><br/>

<div><b>hereby </b>
<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>4</sub></font>();<br/><b>let </b><font color="Maroon">a1</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="E1:57_1_8"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/><b>thus </b><font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="yellow20.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i>, </i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>(), <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><b>let </b><font color="Maroon">b1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c1</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="E17:57_1"/><i><font color="Green">E152</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</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">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">c1</font>;<br/>



<b>assume </b><a NAME="E18:57_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 ;<br/>

<a NAME="E19:57_1"/><i><font color="Green">E154</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">c1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &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">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E20:57_1"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span></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> 
 
<b>by </b><i><a class="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K5" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, , <a class="txt" href="yellow20.html#E1:57"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">b1</font>,<font color="Maroon">c1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, , <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
;<br/>


</div>
