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<b>let </b><font color="Maroon">C</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2" target="_self">transitive</a> <a href="altcat_1.html#V11" target="_self">associative</a>  <a href="altcat_1.html#L2" target="_self">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o4</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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>and </b><br/><a NAME="E2:53"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</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>and </b><br/><a NAME="E3:53"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o3</font>,<font color="Maroon">o4</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> 
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o3</font>,<font color="Maroon">o4</font>;<br/>





<a NAME="E4:53"/><i><font color="Green">E29</font></i>: 
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> is <a href="altcat_1.html#V3" target="_self">associative</a>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E3:53"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E5:53"/><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">o1</font>,<font color="Maroon">o3</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>, , </i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</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="txt" href="altcat_1.html#E5:53"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E8:53"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c</font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font>,<font color="Maroon">o4</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="txt" href="altcat_1.html#E5:53"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E9:53"/>
( <font color="Maroon">a</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">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> &amp; <font color="Maroon">b</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">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> &amp; <font color="Maroon">c</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">o3</font>,<font color="Maroon">o4</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> )
 
<b>by </b><i>, , <a class="txt" href="altcat_1.html#E5:53"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E10:53"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o3</font>,<font color="Maroon">o4</font> )
 
;<br/>
<b>thus </b><font color="Maroon">c</font> <a href="altcat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o3</font>,<font color="Maroon">o4</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E5:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="altcat_1.html#T8" target="_self">Th8</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</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o4</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font>,<font color="Maroon">o4</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, </i>
;<br/>


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