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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V18">full</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V17">faithful</a> )
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E21</font></i>: 
<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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green">E22</font></i>: 
<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>4</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/>

<b>assume </b><a NAME="E4:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="altcat_3.html#V1">retraction</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:33"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E6:33"><i><font color="Green">E23</font></i></a>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E8:33"/>
 <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:33"><i><font color="Green">E20</font></i></a>, <a class="ref" href="altcat_4.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E9:33"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<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">c<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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E10:33"/><i><font color="Green">E26</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">c<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">c<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="txt" href="altcat_4.html#E3:33"><i><font color="Green">E22</font></i></a>, <a class="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:33"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E13:33"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E9:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:33"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</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>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<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>9</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>4</sub></font> <b>by </b><i><a class="txt" href="altcat_4.html#E10:33"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_4.html#E12:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D3">ALTCAT_3:def 3</a><br/>

<a NAME="E16:33"/><i><font color="Green">E30</font></i>: 
 <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:33"><i><font color="Green">E20</font></i></a>, <a class="ref" href="altcat_4.html#T15" target="_self">Th15</a></i>;<br/>
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="altcat_4.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_4.html#E3:33"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_4.html#E7:33"><i><font color="Green">E24</font></i></a>, <a class="txt" href="altcat_4.html#E10:33"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_4.html#E13:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:33"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_4.html#E3:33"><i><font color="Green">E22</font></i></a>, <a class="ref" href="functor0.html#D25">FUNCTOR0:def 25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
;<br/>
<b>hence </b><a NAME="E18:33"/>
<font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E14:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_4.html#E16:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div>
