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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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>, <font color="Maroon">c<sub>4</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="E1:29"/><i><font color="Green">E23</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>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>  &amp; <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>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/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E24</font></i>: 
( <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>  &amp; <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>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>2</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>  )
 
<b>by </b><i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</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>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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<b>assume </b><a NAME="E3:29"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="altcat_3.html#V2" target="_self">coretraction</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="altcat_3.html#V2" target="_self">coretraction</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</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><br/><a NAME="E5:29"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:29"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E3:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_3.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:29"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<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" target="_self">ALTCAT_3:def 3</a><br/>

<a NAME="E10:29"/><i><font color="Green">E29</font></i>: 
<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>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> 
 
<b>by </b><i><a class="ref" href="altcat_1.html#T23">ALTCAT_1:23</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E9:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E2:29"><i><font color="Green">E24</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E7:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="altcat_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E2:29"><i><font color="Green">E24</font></i></a>, <a class="txt" href="altcat_3.html#E10:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</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="altcat_3.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="altcat_3.html#D1" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E12:29"/>
<font color="Maroon">c<sub>9</sub></font> <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D1" target="_self">Def1</a></i>;<br/>


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