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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</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">  for <font color="Olive" title="b1">E1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">B</font><br/>  for <font color="Olive" title="b2">E2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">E1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">E2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b1">E1</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b2">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds <br/><font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">E1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">B</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">  for <font color="Olive" title="b1">E2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c5">E1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">E2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c5">E1</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b1">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds <br/><font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">E2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">B</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="c5">E1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">E2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c5">E1</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:135"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c5">E1</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"> (  not <font color="Maroon" title="c6">E2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  or  not <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c5">E1</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> or  not <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:135"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
<font color="Maroon" title="c6">E2</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"> (  not <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c5">E1</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> or  not <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:135"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c5">E1</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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"> (  not <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c3">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:135"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
<font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">E2</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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">C1</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">C2</font></span><br/>

<a NAME="E5:135"/>
 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="c3">C1</font>,<font color="Maroon" title="c4">C2</font> ex <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="c4">C2</font>,<font color="Maroon" title="c3">C1</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; <font color="Olive" title="b2">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">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="c3">C1</font> &amp; <font color="Olive" title="b1">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">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="c4">C2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="135_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c4">C2</font><span class="kw"> such that </span><br/><a NAME="E2:135_1"/><span class="lab"><font color="Green" title="E83">A5</font></span>: 
( <font color="Maroon" title="c7">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c4">C2</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E4:135"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="cat_8.html#D34" target="_self" onmouseover="rs('cat_8/D34')" onmouseout="rh()">Def34</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c3">C1</font><span class="kw"> such that </span><br/><a NAME="E4:135_1"/><span class="lab"><font color="Green" title="E84">A6</font></span>: 
( <font color="Maroon" title="c8">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c3">C1</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c8">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="ref" href="cat_8.html#D34" target="_self" onmouseover="rs('cat_8/D34')" onmouseout="rh()">Def34</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">F</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">  ex <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="c4">C2</font>,<font color="Maroon" title="c3">C1</font> st <br/>( <font color="Maroon" title="c7">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">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="c3">C1</font> &amp; <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">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="c4">C2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">G</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="c7">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">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="c3">C1</font> &amp; <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">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="c4">C2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:135_1"/>
( <font color="Maroon" title="c7">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></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"> ( <font color="Maroon" title="c8">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">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="c3">C1</font> &amp; <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">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="c4">C2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">H2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c3">C1</font><span class="kw"> such that </span><br/><a NAME="E7:135_1"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
( <font color="Maroon" title="c9">H2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H2</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c3">C1</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">E1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c9">H2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="ref" href="cat_8.html#D34" target="_self" onmouseover="rs('cat_8/D34')" onmouseout="rh()">Def34</a></span>;<br/>
<a NAME="E8:135_1"/><span class="lab"><font color="Green" title="E86">A8</font></span>: 
<font color="Maroon" title="c8">G</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<a NAME="E9:135_1"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<font color="Maroon" title="c5">E1</font> = 
<font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E8:135_1"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="txt" href="#E9:135_1"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E79">A1</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="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="cat_8.html#T50" target="_self" onmouseover="rs('cat_8/T50')" onmouseout="rh()">Th50</a></span>
<br/>.= 
<font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T11" onmouseover="rs('cat_7/T11')" onmouseout="rh()">CAT_7:11</a></span>
;<br/>
<a NAME="E11:135_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A10</font></span>: 
<font color="Maroon" title="c8">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:135_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<font color="Maroon" title="c5">E1</font> = 
<font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="cat_7.html#T11" onmouseover="rs('cat_7/T11')" onmouseout="rh()">CAT_7:11</a></span>
<br/>.= 
<font color="Maroon" title="c5">E1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c3">C1</font></span>)</span> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T51" target="_self" onmouseover="rs('cat_8/T51')" onmouseout="rh()">Th51</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:135_1"/>
<font color="Maroon" title="c8">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c7">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="c3">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:135_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E11:135_1"><span class="lab"><font color="Green" title="E88">A10</font></span></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"> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">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="c4">C2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">H3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c4">C2</font><span class="kw"> such that </span><br/><a NAME="E15:135_1"/><span class="lab"><font color="Green" title="E89">A11</font></span>: 
( <font color="Maroon" title="c10">H3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H3</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c4">C2</font>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">E2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c10">H3</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E4:135"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="cat_8.html#D34" target="_self" onmouseover="rs('cat_8/D34')" onmouseout="rh()">Def34</a></span>;<br/>
<font color="Maroon" title="c6">E2</font> = 
<font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E8:135_1"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="txt" href="#E9:135_1"><span class="lab"><font color="Green" title="E87">A9</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="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="cat_8.html#T50" target="_self" onmouseover="rs('cat_8/T50')" onmouseout="rh()">Th50</a></span>
<br/>.= 
<font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T11" onmouseover="rs('cat_7/T11')" onmouseout="rh()">CAT_7:11</a></span>
;<br/>
<a NAME="E17:135_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A12</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">H3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:135_1"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<font color="Maroon" title="c6">E2</font> = 
<font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="cat_7.html#T11" onmouseover="rs('cat_7/T11')" onmouseout="rh()">CAT_7:11</a></span>
<br/>.= 
<font color="Maroon" title="c6">E2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T51" target="_self" onmouseover="rs('cat_8/T51')" onmouseout="rh()">Th51</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:135_1"/>
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c8">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="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:135_1"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="txt" href="#E17:135_1"><span class="lab"><font color="Green" title="E90">A12</font></span></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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:135"/>
<font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c4">C2</font> <a href="cat_6.html#R2" title="CAT_6:pred.2">are_isomorphic</a> 
 <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>; <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>
