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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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">P1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">C1</font><br/>  for <font color="Olive" title="b2">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">C2</font><br/>  for <font color="Olive" title="b3">Q1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C1</font><br/>  for <font color="Olive" title="b4">Q2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">Q1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">Q2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b3">Q1</font>,<font color="Olive" title="b4">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">P1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">C1</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">P2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">C2</font><br/>  for <font color="Olive" title="b2">Q1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C1</font><br/>  for <font color="Olive" title="b3">Q2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Maroon" title="c5">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">Q1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">Q2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Olive" title="b1">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b2">Q1</font>,<font color="Olive" title="b3">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<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">  for <font color="Olive" title="b1">Q1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C1</font><br/>  for <font color="Olive" title="b2">Q2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Maroon" title="c5">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">Q1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">Q2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">Q1</font>,<font color="Olive" title="b2">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">Q1</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C1</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">Q2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Maroon" title="c5">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">Q1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">Q2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">Q1</font>,<font color="Olive" title="b1">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">Q2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<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="c5">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">Q1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">Q2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c8">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> implies <font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
( <font color="Maroon" title="c5">P1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">P2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">Q1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">Q2</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">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> or  not <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c8">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> or <font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:70"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<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"> (  not <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c8">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> or <font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:70"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c8">Q2</font> <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<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">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E4:70"/>
 ex <font color="Olive" title="b1">FF</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> ex <font color="Olive" title="b2">GG</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">A</font> st <br/>( <font color="Olive" title="b1">FF</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">GG</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">FF</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">A</font> &amp; <font color="Olive" title="b1">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b2">GG</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">B</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c9">FF</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font><span class="kw"> such that </span><br/><a NAME="E2:70_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
( <font color="Maroon" title="c9">FF</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">P1</font> &amp; <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P2</font> &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">A</font>,<font color="Maroon" title="c4">B</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="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">P1</font> &amp; <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P2</font> holds <br/><font color="Maroon" title="c9">FF</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="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="cat_8.html#D17" target="_self" onmouseover="rs('cat_8/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">GG</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">A</font><span class="kw"> such that </span><br/><a NAME="E4:70_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
( <font color="Maroon" title="c10">GG</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">Q1</font> &amp; <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">Q2</font> &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">B</font>,<font color="Maroon" title="c3">A</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">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">Q1</font> &amp; <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">Q2</font> holds <br/><font color="Maroon" title="c10">GG</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="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="cat_8.html#D17" target="_self" onmouseover="rs('cat_8/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">FF</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">GG</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">A</font> st <br/>( <font color="Maroon" title="c9">FF</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">GG</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</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">A</font> &amp; <font color="Maroon" title="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">GG</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">B</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">GG</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">FF</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c10">GG</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c10">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</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">A</font> &amp; <font color="Maroon" title="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</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">B</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:70_1"/>
( <font color="Maroon" title="c9">FF</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c10">GG</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="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</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="c10">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</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">A</font> &amp; <font color="Maroon" title="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</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">B</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">G11</font> = <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">G12</font> = <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c3">A</font><span class="kw"> such that </span><br/><a NAME="E7:70_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
( <font color="Maroon" title="c13">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> &amp; <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c13">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> &amp; (  for <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="c3">A</font>,<font color="Maroon" title="c3">A</font>  st <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="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> &amp; <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> holds <br/><font color="Maroon" title="c13">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="cat_8.html#D17" target="_self" onmouseover="rs('cat_8/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<a NAME="E8:70_1"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_7.html#T10" onmouseover="rs('cat_7/T10')" onmouseout="rh()">CAT_7:10</a></span>;<br/>
<a NAME="E9:70_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_7.html#T10" onmouseover="rs('cat_7/T10')" onmouseout="rh()">CAT_7:10</a></span>;<br/>
<a NAME="E10:70_1"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c5">P1</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> <font color="Maroon" title="c3">A</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">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 NAME="E11:70_1"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
<font color="Maroon" title="c6">P2</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> <font color="Maroon" title="c3">A</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">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/>
<span class="kw">thus </span><font color="Maroon" title="c10">GG</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">FF</font> = 
<font color="Maroon" title="c13">H1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E8:70_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E9:70_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>
<br/>.= 
 <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c3">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E10:70_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E11:70_1"><span class="lab"><font color="Green" title="E47">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="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</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">B</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">G21</font> = <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">G22</font> = <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">H2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c4">B</font><span class="kw"> such that </span><br/><a NAME="E14:70_1"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
( <font color="Maroon" title="c16">H2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">H2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> &amp; <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c16">H2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> &amp; (  for <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="c4">B</font>,<font color="Maroon" title="c4">B</font>  st <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">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> &amp; <font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> holds <br/><font color="Maroon" title="c16">H2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="cat_8.html#D17" target="_self" onmouseover="rs('cat_8/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<a NAME="E15:70_1"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<font color="Maroon" title="c7">Q1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_7.html#T10" onmouseover="rs('cat_7/T10')" onmouseout="rh()">CAT_7:10</a></span>;<br/>
<a NAME="E16:70_1"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
<font color="Maroon" title="c8">Q2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_7.html#T10" onmouseover="rs('cat_7/T10')" onmouseout="rh()">CAT_7:10</a></span>;<br/>
<a NAME="E17:70_1"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c7">Q1</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> <font color="Maroon" title="c4">B</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">P1</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</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 NAME="E18:70_1"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<font color="Maroon" title="c8">Q2</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> <font color="Maroon" title="c4">B</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P2</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_7.html#T11" onmouseover="rs('cat_7/T11')" onmouseout="rh()">CAT_7:11</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c9">FF</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c10">GG</font> = 
<font color="Maroon" title="c16">H2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E15:70_1"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E16:70_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>
<br/>.= 
 <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c4">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E17:70_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="txt" href="#E18:70_1"><span class="lab"><font color="Green" title="E52">A15</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="E5:70"/>
<font color="Maroon" title="c3">A</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c4">B</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>; <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>
