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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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">C</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">F</font> being <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">C</font>  st <font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</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="b7">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font><br/> for <font color="Olive" title="b8">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font>  st <font color="Olive" title="b8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b7">f</font></span><br/><b>let </b><font color="Maroon" title="c3">C</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font>; <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="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</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="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font><br/> for <font color="Olive" title="b7">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font>  st <font color="Olive" title="b7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b6">f</font></span><br/><b>let </b><font color="Maroon" title="c4">F</font> be  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <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">A</font><br/> for <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</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="b5">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><br/> for <font color="Olive" title="b6">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>  st <font color="Olive" title="b6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">f</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <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">c</font>, <font color="Olive" title="b2">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font>  st <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b3">f</font></span><br/><b>let </b><font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</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="c5">a</font>,<font color="Maroon" title="c6">b</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font> )</span><br/>









<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</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="c5">a</font>,<font color="Maroon" title="c6">b</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font></span><br/>

<a NAME="E2:42"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="yellow20.html#K3" title="YELLOW20:func.3">incl</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default"><a href="yellow20.html#K3" title="YELLOW20:func.3">incl</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T28" title="FUNCTOR0:th.28">FUNCTOR0:28</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>; <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">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font></span><br/><b>let </b><font color="Maroon" title="c10">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font> )</span><br/>



<b>assume </b><a NAME="E3:42"/>
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font></span><br/>

<a NAME="E4:42"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow20.html#K3" title="YELLOW20:func.3">incl</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="yellow20.html#T28" target="_self" title="YELLOW20:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E5:42"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="yellow20.html#K5" title="YELLOW20:func.5">|</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E2:42"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="functor3.html#T8" title="FUNCTOR3:th.8">FUNCTOR3:8</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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