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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="yellow18.html#V4" title="YELLOW18:attr.4">concrete</a> <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">a</font>, <font color="Olive" title="b2">b</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">a</font>,<font color="Olive" title="b2">b</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">b</font>,<font color="Olive" title="b1">a</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">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">f</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b2">b</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c2">a</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">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">f</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font> )</span><br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E38">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">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c2">a</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">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">f</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font></span><br/>

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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">f</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font><b> such that </b><a NAME="E3:81"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c4">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D2" title="ALTCAT_3:def.2">ALTCAT_3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E4:81"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">g</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="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E5:81"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self" title="YELLOW18:th.38">Th38</a></i>;<br/>
<a NAME="E6:81"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c4">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c2">a</font>, <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font>, <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="yellow18.html#T35" target="_self" title="YELLOW18:th.35">Th35</a></i>;<br/>
<a NAME="E7:81"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow18.html#D10" target="_self" title="YELLOW18:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E8:81"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="#E4:81"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="#E5:81"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="#E6:81"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="funct_2.html#T24" title="FUNCT_2:th.24">FUNCT_2:24</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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