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<b>let </b><font color="Maroon">C</font> be  <a href="altcat_1.html#V9">semi-functional</a> <a href="yellow18.html#V2" target="_self">para-functional</a> <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><a NAME="E1:84"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<b>assume </b><a NAME="E2:84"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>assume </b><a NAME="E3:84"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font> ;<br/>
<a NAME="E5:84"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:84"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E6:84"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:84"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E7:84"/><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:84"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>;<br/>
<a NAME="E8:84"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:84"/><b>then </b>
( <font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="yellow18.html#E7:84"><i><font color="Green">E43</font></i></a>, , , <a class="txt" href="yellow18.html#E1:84"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E10:84"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g</font> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:84"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E11:84"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">g</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E12:84"/><b>then </b>
( <font color="Maroon">f</font> is <a href="altcat_3.html#V1">retraction</a> &amp; <font color="Maroon">f</font> is <a href="altcat_3.html#V2">coretraction</a> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a>, <a class="ref" href="altcat_3.html#D3">ALTCAT_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:84"/>
( <font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="txt" href="yellow18.html#E7:84"><i><font color="Green">E43</font></i></a>, , , <a class="ref" href="altcat_3.html#D4">ALTCAT_3:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a><br/>


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