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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</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>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:81"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">c<sub>4</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a><br/>

<a NAME="E3:81"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:81"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E4:81"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:81"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E5:81"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>2</sub></font>, <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>3</sub></font>, <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:81"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow18.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E6:81"/>
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow18.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E7:81"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:81"><i><font color="Green">E40</font></i></a>, <a class="txt" href="yellow18.html#E4:81"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yellow18.html#E5:81"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#T24">FUNCT_2:24</a></i>;<br/>


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