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<div class="add">

<b>thus </b><a NAME="E1:56_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#D4" target="_self">YELLOW20:def 4</a><br/>

<a NAME="E2:56_1"/><i><font color="Green">E35</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#K4" 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#K4" 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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" 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><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#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><a NAME="E1:56_1_1"/>
<font color="Maroon">c<sub>1</sub></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<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></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="txt" href="yellow20.html#E2:56_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E3:56_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>1</sub></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<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self">Th29</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></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#E2:56"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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="E1:56_1_2"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E2:56_1_2"/>
( <font color="Maroon">c<sub>1</sub></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<sub>2</sub></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<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></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="txt" href="yellow20.html#E2:56_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></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="txt" href="yellow20.html#E3:56_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E5:56_1_2"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>7</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ;<br/><a NAME="E7:56_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>1</sub></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<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></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<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow20.html#T29" target="_self">Th29</a>, <a class="ref" href="yellow20.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E8:56_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>7</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E3:56"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow20.html#E5:56_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E9:56_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:56_1_3"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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="E2:56_1_3"/>
( <font color="Maroon">c<sub>1</sub></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<sub>2</sub></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<sub>3</sub></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<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></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="txt" href="yellow20.html#E2:56_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></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="E4:56_1_3"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) )
 ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/><b>assume </b><a NAME="E5:56_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>) )
 ;<br/><a NAME="E6:56_1_3"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>7</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_3"><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">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> ;<br/><a NAME="E8:56_1_3"/><i><font color="Green">E43</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>8</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_3"><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">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> ;<br/><a NAME="E10:56_1_3"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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#E1:56_1_3"><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="E11:56_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>;<br/><a NAME="E12:56_1_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></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<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E4:56_1_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow20.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E13:56_1_3"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yellow20.html#E5:56_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yellow20.html#E10:56_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="yellow20.html#E11:56_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E14:56_1_3"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>12</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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>13</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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E3:56"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow20.html#E6:56_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow20.html#E8:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="yellow20.html#E12:56_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font>) = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>15</sub></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#K6">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow20.html#E6:56_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow20.html#E8:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="yellow20.html#E13:56_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="functor0.html#D24">FUNCTOR0:def 24</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E12:56_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="yellow20.html#E13:56_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yellow20.html#E14:56_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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="E1:56_1_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 ;<br/><a NAME="E2:56_1_4"/>
<font color="Maroon">c<sub>1</sub></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<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></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="txt" href="yellow20.html#E2:56_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E4:56_1_4"/>
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>1</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>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>1</sub></font>) = 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow20.html#T30" target="_self">Th30</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">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor2.html#T2">FUNCTOR2:2</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#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self">Th29</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_4"><i><font color="Green">E40</font></i></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">c<sub>1</sub></font> being  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</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="E8:56_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>() holds <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E9:56_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</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="yellow18.html#S8">YELLOW18:sch 8</a>(<a class="txt" href="yellow20.html#E3:56_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="yellow20.html#E4:56_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="yellow20.html#E5:56_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow20.html#E6:56_1"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<a NAME="E10:56_1"/><i><font color="Green">E42</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#E1:56"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E11:56_1"/><b>then </b><i><font color="Green">E43</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">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><a NAME="E1:56_1_5"/>
( <font color="Maroon">c<sub>2</sub></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<sub>3</sub></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<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></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="txt" href="yellow20.html#E2:56_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>assume </b><a NAME="E3:56_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 ;<br/><b>then </b><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self">Th29</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self">Th29</a></i>
;<br/><b>hence </b><a NAME="E5:56_1_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow20.html#E11:56_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow20.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:56_1_6"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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="E2:56_1_6"/>
( <font color="Maroon">c<sub>2</sub></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<sub>3</sub></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<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></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="txt" href="yellow20.html#E2:56_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E4:56_1_6"/><i><font color="Green">E47</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>7</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> ;<br/><b>assume </b><a NAME="E6:56_1_6"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>)
 ;<br/><b>then </b><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self">Th30</a></i>
;<br/><b>hence </b><a NAME="E8:56_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow20.html#E11:56_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="yellow20.html#E4:56_1_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow20.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:56_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>4</sub></font>()ex <font color="Olive">b<sub>6</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>5</sub></font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:56_1_7"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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="E2:56_1_7"/>
( <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E4:56_1_7"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_7"><i><font color="Green">E47</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>10</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E7:56_1_7"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></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<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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<sub>9</sub></font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E10:56_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow20.html#E4:56_1_7"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow20.html#T34" target="_self">Th34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></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#E2:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow20.html#E7:56_1_7"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E9:56_1_7"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow20.html#E3:56"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow20.html#E1:56_1_7"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yellow20.html#E7:56_1_7"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="yellow20.html#E3:56"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow20.html#E1:56_1_7"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yellow20.html#E7:56_1_7"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E11:56_1_7"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>11</sub></font>) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow20.html#E7:56_1_7"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow20.html#E9:56_1_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow20.html#T29" target="_self">Th29</a>, <a class="ref" href="yellow20.html#T30" target="_self">Th30</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E15:56_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="functor0.html#V21">bijective</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S10">YELLOW18:sch 10</a>(<a class="txt" href="yellow20.html#E8:56_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="yellow20.html#E9:56_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yellow20.html#E12:56_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="yellow20.html#E13:56_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow20.html#E14:56_1"><i><font color="Green">E46</font></i></a>);</i><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="E1:56_1_8"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E8:56_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E1:56_1_8"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow20.html#T29" target="_self">Th29</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">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">F<sub>1</sub></font>();<br/>



<b>assume </b><a NAME="E17:56_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>



<b>assume </b><a NAME="E18:56_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E19:56_1"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E17:56_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E20:56_1"/><b>then </b><i><font color="Green">E50</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">c<sub>4</sub></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">c<sub>5</sub></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#D19">FUNCTOR0:def 19</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" target="_self">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E9:56_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yellow20.html#E17:56_1"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E17:56_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yellow20.html#E18:56_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E19:56_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow20.html#E20:56_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>
;<br/>


</div>
