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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E11</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E12</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="cat_1.html#M1">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>6</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<a NAME="E3:25"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E1:25"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E2:25"><i><font color="Green">E12</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>then </b><i><font color="Green">E15</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_2.html#E3:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
;<br/>
<a NAME="E6:25"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E3:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="altcat_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:25"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:25"><i><font color="Green">E14</font></i></a>, <a class="ref" href="altcat_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E8:25"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E6:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="altcat_2.html#D4" target="_self">Def4</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="altcat_2.html#E8:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_2.html#E5:25"><i><font color="Green">E15</font></i></a>, <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_2.html#E1:25"><i><font color="Green">E11</font></i></a>, <a class="txt" href="altcat_2.html#E2:25"><i><font color="Green">E12</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/>


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