<?xml version="1.0"?>
<div class="add">

<a NAME="E1:10_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_1.html#D22">CAT_1:def 22</a></i>;<br/>
<a NAME="E2:10_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:10_1_1"/><i><font color="Green">E14</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_1.html#D22">CAT_1:def 22</a></i>;<br/>
<a NAME="E4:10_1_1"/><i><font color="Green">E15</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E3:10_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E6:10_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D1" target="_self">ISOCAT_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:10_1_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="isocat_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:10_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:10_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:10_1_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="isocat_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:10_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="isocat_1.html#E2:10_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:10_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="isocat_1.html#E2:10_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:10_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="isocat_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:10_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="isocat_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E7:10_1_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E8:10_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>

<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="isocat_1.html#E6:10_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="isocat_1.html#E8:10_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E4:10_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>

;<br/>
<a NAME="E10:10_1_1_1"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E2:10_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="isocat_1.html#E7:10_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E4:10_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="isocat_1.html#E6:10_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="isocat_1.html#E4:10_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="isocat_1.html#E6:10_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:10_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a>  is    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
