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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">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>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">o1</font> is <a href="altcat_3.html#V7" target="_self">terminal</a>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">o2</font> is <a href="altcat_3.html#V7" target="_self">terminal</a>
 ;<br/>

<b>consider </b><font color="Maroon">M1</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font><b> such that </b><br/><a NAME="E4:47"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">M1</font> <a href="hidden.html#R2">in</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">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:47"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">M2</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o1</font><b> such that </b><br/><a NAME="E6:47"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">M2</font> <a href="hidden.html#R2">in</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> &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> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:47"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E7:47"/>
( <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>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D6" target="_self">ALTCAT_3:def 6</a><br/>

<a NAME="E8:47"/><i><font color="Green">E28</font></i>: 
(  ex <font color="Olive">M</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o1</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o1</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">o1</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> ) &amp;  ex <font color="Olive">N</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o2</font> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</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">o2</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> ) )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E6:47"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E8:47"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:47"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:47"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E13:47"/>
( <font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E12:47"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:47"/><b>then </b>
( <font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E12:47"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:47"/><b>then </b>
( <font color="Maroon">M2</font> <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">M2</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">M1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E16:47"/><b>then </b>
( <font color="Maroon">M1</font> is <a href="altcat_3.html#V1" target="_self">retraction</a> &amp; <font color="Maroon">M1</font> is <a href="altcat_3.html#V2" target="_self">coretraction</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:47"/><b>then </b>
<font color="Maroon">M1</font> is <a href="altcat_3.html#V3" target="_self">iso</a>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E18:47"/>
 ex <font color="Olive">A</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> st <font color="Olive">A</font> is <a href="altcat_3.html#V3" target="_self">iso</a>
 ;<br/>


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