<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C1</font> be   <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <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">C2</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Olive" title="b1">C2</font>  st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_8.html#K3" title="CAT_8:func.3">OrdC0-&gt;</a> <font color="Olive" title="b1">C2</font> holds <br/> <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Olive" title="b2">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Olive" title="b1">C2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_8.html#K3" title="CAT_8:func.3">OrdC0-&gt;</a> <font color="Maroon" title="c2">C2</font> holds <br/> <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Olive" title="b1">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">E</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</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">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_8.html#K3" title="CAT_8:func.3">OrdC0-&gt;</a> <font color="Maroon" title="c2">C2</font> implies  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Maroon" title="c3">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:130"/>
<font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_8.html#K3" title="CAT_8:func.3">OrdC0-&gt;</a> <font color="Maroon" title="c2">C2</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">  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Maroon" title="c3">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">C</font> =  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">E</font> = <font color="Maroon" title="c3">E</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> ;<br/>
<a NAME="E3:130"/>
 for <font color="Olive" title="b1">D</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">D</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b2">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> st <br/>( <font color="Olive" title="b3">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H</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">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  st <font color="Olive" title="b4">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">H1</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="130_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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">  for <font color="Olive" title="b1">F</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="c6">D</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b2">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> st <br/>( <font color="Olive" title="b2">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H</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">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b3">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  st <font color="Olive" title="b3">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">H1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</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="c6">D</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</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>  implies  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</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">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:130_1"/>
<font color="Maroon" title="c7">F</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">  ex <font color="Olive" title="b1">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</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">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  st <font color="Olive" title="b2">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">H1</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">H</font> = <font color="Maroon" title="c6">D</font> <a href="cat_8.html#K2" title="CAT_8:func.2">-&gt;OrdC1</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">H</font> = <font color="Maroon" title="c6">D</font> <a href="cat_8.html#K2" title="CAT_8:func.2">-&gt;OrdC1</a>  as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">H</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="c9">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</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">C1</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="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  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="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</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">C1</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:130_1"/>
<font color="Maroon" title="c9">H</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="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</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">C1</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="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  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="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</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">C1</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:130_1"/>
<font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</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">C1</font></span>)</span></span>)</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">  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="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  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="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</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">C1</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">H1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</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="c10">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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">C1</font></span>)</span></span>)</span> implies <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">H1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:130_1"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
<font color="Maroon" title="c10">H1</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="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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">C1</font></span>)</span></span>)</span> or <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">H1</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">H2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:130_1"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
( <font color="Maroon" title="c11">H2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b1">H3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>  st <font color="Olive" title="b1">H3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/><font color="Maroon" title="c11">H2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H3</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D4" target="_self" onmouseover="rs('cat_8/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:130_1"/>
<font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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">C1</font></span>)</span></span>)</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="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">H1</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c9">H</font> = 
<font color="Maroon" title="c11">H2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:130_1"><span class="lab"><font color="Green" title="E77">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">H1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:130_1"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E5:130_1"><span class="lab"><font color="Green" title="E76">A1</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="E4:130"/>
 <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Maroon" title="c3">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D34" target="_self" onmouseover="rs('cat_8/D34')" onmouseout="rh()">Def34</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>
