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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">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="Olive" title="b2">o2</font>,<font color="Olive" title="b1">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>  holds <br/>for <font color="Olive" title="b3">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>  st <font color="Olive" title="b3">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> holds <br/><font color="Olive" title="b3">A</font> is <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  implies for <font color="Olive" title="b1">A</font> being  <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>  st <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> holds <br/><font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> )</span><br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E20">A1</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <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>  st <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> holds <br/><font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> )</span><br/>

<b>assume </b><a NAME="E2:25"/>
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">R</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="E4:25"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c5">R</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="ref" href="altcat_3.html#D2" target="_self" title="ALTCAT_3:def.2">Def2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D8" target="_self" title="ALTCAT_3:def.8">ALTCAT_3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c6">o</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>  implies for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</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="c6">o</font>  st <font color="Olive" title="b1">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> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> )</span><br/>

<b>assume </b><a NAME="E5:25"/><i><font color="Green" title="E22">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="c3">o2</font>,<font color="Maroon" title="c6">o</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</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="c6">o</font>  st <font color="Olive" title="b1">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> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font></span><br/>

<b>let </b><font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <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="c6">o</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">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> <font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> )</span><br/>

<b>assume </b><a NAME="E6:25"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c7">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> <font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font></span><br/>

<b>thus </b><font color="Maroon" title="c7">B</font> = 
<font color="Maroon" title="c7">B</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="c3">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:25"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c7">B</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_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">R</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E4:25"><i><font color="Green" title="E21">A2</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="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">R</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:25"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="altcat_3.html#E5:25"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="altcat_3.html#E6:25"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon" title="c8">C</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_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">R</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:25"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="altcat_3.html#E5:25"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon" title="c8">C</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="c3">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E4:25"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:25"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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