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<b>let </b><font color="Maroon">C</font> be  <a href="yellow18.html#V4" target="_self">concrete</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="E1:82"/><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>  &amp; <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> <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>given </b><font color="Maroon">g</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font><b> such that </b><a NAME="E2:82"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">f</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="E3:82"/><i><font color="Green">E39</font></i>: 
<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="altcat_1.html#K8">idm</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E4:82"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</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#E3:82"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yellow18.html#E1:82"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E5:82"/><i><font color="Green">E43</font></i>: 
 <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><a class="txt" href="yellow18.html#E3:82"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E6:82"/>
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</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/>;<br/>
<b>hence </b><a NAME="E7:82"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="txt" href="yellow18.html#E4:82"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E5:82"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T53">FUNCT_1:53</a></i>;<br/>


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