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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> is  <a href="cat_8.html#V3" title="CAT_8:attr.3">terminal</a>  &amp; <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font> implies <font color="Maroon" title="c2">D</font> is  <a href="cat_8.html#V3" title="CAT_8:attr.3">terminal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">C</font> is  <a href="cat_8.html#V3" title="CAT_8:attr.3">terminal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font> or <font color="Maroon" title="c2">D</font> is  <a href="cat_8.html#V3" title="CAT_8:attr.3">terminal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:33"/>
<font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">D</font> is  <a href="cat_8.html#V3" title="CAT_8:attr.3">terminal</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c4">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E4:33"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c4">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D28" onmouseover="rs('cat_6/D28')" onmouseout="rh()">CAT_6:def 28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">B</font> be   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_8.html#D4" target="_self" onmouseover="rs('cat_8/D4')" onmouseout="rh()">CAT_8:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c2">D</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b2">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b2">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/><font color="Olive" title="b1">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">G</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">F1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
( <font color="Maroon" title="c6">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b1">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/><font color="Maroon" title="c6">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">F2</font> = <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b1">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G</font> ) )</span><br/>

<a NAME="E7:33"/>
 for <font color="Olive" title="b1">G1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">G1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  implies <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">G1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c8">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">G1</font></span><br/>

<span class="kw">hence </span><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> = 
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="cat_7.html#T10" onmouseover="rs('cat_7/T10')" onmouseout="rh()">CAT_7:10</a></span>
<br/>.= 
<font color="Maroon" title="c8">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="cat_7.html#T11" onmouseover="rs('cat_7/T11')" onmouseout="rh()">CAT_7:11</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:33"/>
( <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b1">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c6">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">G</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
