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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="yellow18.html#V4" title="YELLOW18:attr.4">concrete</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E39">A1</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>

<b>given </b><font color="Maroon" title="c5">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font><b> such that </b><a NAME="E2:82"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D3" title="ALTCAT_3:def.3">ALTCAT_3:def 3</a><br/>

<a NAME="E3:82"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:82"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E4:82"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:82"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self" title="YELLOW18:th.38">Th38</a></i>;<br/>
<a NAME="E5:82"/><i><font color="Green" title="E43">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:82"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="yellow18.html#T36" target="_self" title="YELLOW18:th.36">Th36</a></i>;<br/>
<a NAME="E6:82"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow18.html#D10" target="_self" title="YELLOW18:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E7:82"/>
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:82"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="yellow18.html#E4:82"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="yellow18.html#E5:82"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="funct_1.html#T53" title="FUNCT_1:th.53">FUNCT_1:53</a></i>;<br/>


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