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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="altcat_1.html#K1" target="_self">^&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>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="altcat_1.html#K1" target="_self">^&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>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<a NAME="E3:50_1_2"/>
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V5" target="_self">with_left_units</a>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D18" target="_self">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:50_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E7:50_1_2"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&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="altcat_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:50_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E9:50_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E7:50_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E10:50_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E11:50_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E7:50_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E12:50_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E6:50_1_2"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E7:50_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E1:50_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E7:50_1_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:50_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_1.html#E7:50_1_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E7:50_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E6:50_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_1.html#E10:50_1_2"><i><font color="Green">E37</font></i></a></i>
;<br/><br/>


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