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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</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>let </b><font color="Maroon" title="c4">A</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a> )
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E25">A2</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">o1</font>,<font color="Maroon" title="c3">o2</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">o2</font>,<font color="Maroon" title="c2">o1</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/>

<a NAME="E3:30"/><i><font color="Green" title="E26">A3</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">o1</font>,<font color="Maroon" title="c2">o1</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> 
 
<b>by </b><i><a class="ref" href="altcat_1.html#T23" title="ALTCAT_1:th.23">ALTCAT_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">B</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c2">o1</font><b> such that </b><br/><a NAME="E5:30"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="altcat_3.html#D2" target="_self" title="ALTCAT_3:def.2">Def2</a></i>;<br/>
<a NAME="E6:30"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E5:30"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>;<br/>
<a NAME="E8:30"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</a></i>;<br/>
<a NAME="E9:30"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>;<br/>
<a NAME="E10:30"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</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">o1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="altcat_3.html#E3:30"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="altcat_3.html#D7" target="_self" title="ALTCAT_3:def.7">Def7</a></i>;<br/>
<a NAME="E11:30"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></i>;<br/>
<a NAME="E12:30"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>
 
<b>by </b><i><a class="ref" href="altcat_3.html#D3" target="_self" title="ALTCAT_3:def.3">Def3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E31">A8</font></i>: <font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> = 
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="altcat_3.html#E5:30"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="altcat_3.html#E11:30"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="altcat_3.html#D4" target="_self" title="ALTCAT_3:def.4">Def4</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:30"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> = 
<font color="Maroon" title="c5">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="altcat_3.html#E5:30"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="altcat_3.html#E11:30"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="altcat_3.html#E12:30"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="altcat_3.html#D4" target="_self" title="ALTCAT_3:def.4">Def4</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o1</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="altcat_3.html#E3:30"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="altcat_3.html#E9:30"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="altcat_3.html#D7" target="_self" title="ALTCAT_3:def.7">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E15:30"/>
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E13:30"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="altcat_3.html#D5" target="_self" title="ALTCAT_3:def.5">Def5</a></i>;<br/>


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