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<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>, <font color="Maroon">p2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p1</font>,<font color="Maroon">p2</font>;<br/>











<b>assume </b><a NAME="E1:45"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</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">p2</font>,<font color="Maroon">p1</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>thus </b><a NAME="E2:45"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">n</font> is <a href="altcat_3.html#V1">retraction</a> implies <font color="Maroon">m</font> is <a href="altcat_3.html#V1">retraction</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon">n</font> is <a href="altcat_3.html#V1">retraction</a>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p2</font>,<font color="Maroon">p1</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">B</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o1</font> <b>by </b><i>, <a class="ref" href="altcat_2.html#T34">ALTCAT_2:34</a></i>;<br/>
<b>take </b>
<font color="Maroon">B1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a><br/>

<b>thus </b><a NAME="E5:45_1"/>
<font color="Maroon">B1</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="altcat_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:45"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">n</font> is <a href="altcat_3.html#V2">coretraction</a> implies <font color="Maroon">m</font> is <a href="altcat_3.html#V2">coretraction</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon">n</font> is <a href="altcat_3.html#V2">coretraction</a>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p2</font>,<font color="Maroon">p1</font><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">B</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D3">ALTCAT_3:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o1</font> <b>by </b><i>, <a class="ref" href="altcat_2.html#T34">ALTCAT_2:34</a></i>;<br/>
<b>take </b>
<font color="Maroon">B1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D3">ALTCAT_3:def 3</a><br/>

<b>thus </b><a NAME="E5:45_2"/>
<font color="Maroon">B1</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="altcat_4.html#E3:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:45"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> is <a href="altcat_3.html#V3">iso</a>
 ;<br/>

<a NAME="E5:45"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E6:45"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">o2</font>,<font color="Maroon">o1</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="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E7:45"/>
<font color="Maroon">m</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a>, , <a class="txt" href="altcat_4.html#E3:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#T5">ALTCAT_3:5</a>, <a class="ref" href="altcat_3.html#T6">ALTCAT_3:6</a></i>;<br/>


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