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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">C2</font> be   non  <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"> ( <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>) implies  ex <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"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b2">D</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b3">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="b2">D</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">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">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="c2">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="c2">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">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">  ex <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"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  for <font color="Olive" title="b2">D</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b3">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="b2">D</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">C2</font>  st <font color="Olive" title="b3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b4">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b4">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">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">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="c2">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b5">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b5">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="c2">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">H1</font> ) ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> ex <font color="Olive" title="b4">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b5">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">d</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c2</font> &amp; <font color="Olive" title="b4">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">c1</font> &amp; <font color="Olive" title="b5">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  );<br/>
<a NAME="E2:128"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c4">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c7">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E4:128_1"/><span class="lab"><font color="Green" title="E76">A3</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T52" target="_self" onmouseover="rs('cat_8/T52')" onmouseout="rh()">Th52</a></span>;<br/>
<a NAME="E5:128_1"/><span class="lab"><font color="Green" title="E77">A4</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Olive" title="b3">t</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D28" target="_self" onmouseover="rs('cat_8/D28')" onmouseout="rh()">Def28</a></span>;<br/>
<a NAME="E6:128_1"/>
<font color="Maroon" title="c6">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font>
 
;<br/>
<a NAME="E7:128_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Olive" title="b3">t</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_1"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c9">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c10">t</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font><span class="kw"> such that </span><br/><a NAME="E9:128_1"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c9">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c8">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F2</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">F12</font> = <font color="Maroon" title="c10">t</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">c2</font> = <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">y</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="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c13">y</font>] )</span><br/>

<a NAME="E12:128_1"/>
<font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c3">C2</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E13:128_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c13">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">c</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">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> ex <font color="Olive" title="b3">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b4">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c2</font> &amp; <font color="Olive" title="b3">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> &amp; <font color="Olive" title="b4">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">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">  ex <font color="Olive" title="b1">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> ex <font color="Olive" title="b2">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b3">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c2</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Olive" title="b3">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">d</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b2">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c2</font> &amp; <font color="Olive" title="b1">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Olive" title="b2">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">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">  ex <font color="Olive" title="b1">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Olive" title="b1">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">F12</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">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E14:128_1"/>
( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_1"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E4:128_1"><span class="lab"><font color="Green" title="E76">A3</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">consider </span><font color="Maroon" title="c4">E</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E4:128"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E2:128"><span class="lab"><font color="Green" title="E75">A2</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">E</font> = <font color="Maroon" title="c4">E</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">C2</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">E</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">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; (  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="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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> ) ) ) )</span><br/>

<a NAME="E6:128"/><span class="lab"><font color="Green" title="E77">A7</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>  ex <font color="Olive" title="b2">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b4">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b5">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c2</font> &amp; <font color="Olive" title="b4">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">c1</font> &amp; <font color="Olive" title="b5">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></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">  ex <font color="Olive" title="b1">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b3">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b4">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c2</font> &amp; <font color="Olive" title="b3">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">c1</font> &amp; <font color="Olive" title="b4">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">f</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E2:128_2"/>
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>
 
;<br/>
<a NAME="E3:128_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c10">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>, <font color="Maroon" title="c11">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c12">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E5:128_2"/><span class="lab"><font color="Green" title="E78">A8</font></span>: 
( <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font> &amp; <font color="Maroon" title="c5">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c12">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128"><span class="lab"><font color="Green" title="E76">A6</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">c</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">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> ex <font color="Olive" title="b2">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b3">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c2</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> &amp; <font color="Olive" title="b3">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">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">  ex <font color="Olive" title="b1">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b2">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c2</font> &amp; <font color="Olive" title="b1">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Olive" title="b2">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">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">  ex <font color="Olive" title="b1">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Olive" title="b1">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c12">F12</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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c12">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:128_2"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c12">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_2"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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>
<a NAME="E7:128"/>
 for <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>  st <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></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="c6">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_3"/><span class="lab"><font color="Green" title="E78">A9</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c8">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c9">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c10">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:128_3"/><span class="lab"><font color="Green" title="E79">A10</font></span>: 
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c2</font> &amp; <font color="Maroon" title="c9">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">c1</font> &amp; <font color="Maroon" title="c10">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E4:128_3"/><span class="lab"><font color="Green" title="E80">A11</font></span>: 
( <font color="Maroon" title="c7">c</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c8">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_3"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E3:128_3"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="cat_8.html#T56" target="_self" onmouseover="rs('cat_8/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">F</font> being   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E6:128_3"/><span class="lab"><font color="Green" title="E81">A12</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c11">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E4:128_3"><span class="lab"><font color="Green" title="E80">A11</font></span></a>, <a class="ref" href="cat_8.html#T64" target="_self" onmouseover="rs('cat_8/T64')" onmouseout="rh()">Th64</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:128_3"/>
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_3"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="txt" href="#E4:128_3"><span class="lab"><font color="Green" title="E80">A11</font></span></a>, <a class="txt" href="#E6:128_3"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</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>
<a NAME="E8:128"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A13</font></span>: 
<font color="Maroon" title="c5">E</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/>
<a NAME="E9:128"/>
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>  st <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> holds <br/>( <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></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="c6">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">f2</font> implies ( <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_4"/><span class="lab"><font color="Green" title="E79">A14</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">f2</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">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">c11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c10">c12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c11">F11</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:128_4"/><span class="lab"><font color="Green" title="E80">A15</font></span>: 
( <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c9">c11</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c12</font> &amp; <font color="Maroon" title="c10">c12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">c11</font> &amp; <font color="Maroon" title="c11">F11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c1</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c13">c21</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c14">c22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c15">F22</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E5:128_4"/><span class="lab"><font color="Green" title="E81">A16</font></span>: 
( <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">c2</font>,<font color="Maroon" title="c13">c21</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c22</font> &amp; <font color="Maroon" title="c14">c22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">c21</font> &amp; <font color="Maroon" title="c15">F22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c2</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E6:128_4"/><span class="lab"><font color="Green" title="E82">A17</font></span>: 
( <font color="Maroon" title="c8">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">c2</font> &amp; <font color="Maroon" title="c9">c11</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">c21</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_4"><span class="lab"><font color="Green" title="E79">A14</font></span></a>, <a class="txt" href="#E3:128_4"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="txt" href="#E5:128_4"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">F</font>, <font color="Maroon" title="c17">F1</font>, <font color="Maroon" title="c18">F2</font> being   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c19">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c16">F</font>, <font color="Maroon" title="c20">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c16">F</font>,<font color="Maroon" title="c18">F2</font><span class="kw"> such that </span><br/><a NAME="E8:128_4"/><span class="lab"><font color="Green" title="E83">A18</font></span>: 
( <font color="Maroon" title="c8">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c16">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; (  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>  st <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> holds <br/>( <font color="Maroon" title="c20">T2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">T1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">T2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">T1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#T63" target="_self" onmouseover="rs('cat_8/T63')" onmouseout="rh()">Th63</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:128_4"/>
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_4"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="txt" href="#E5:128_4"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="txt" href="#E6:128_4"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="txt" href="#E8:128_4"><span class="lab"><font color="Green" title="E83">A18</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="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c21">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c22">d1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c23">d2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c24">G12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E11:128_4"/><span class="lab"><font color="Green" title="E84">A19</font></span>: 
( <font color="Maroon" title="c6">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c21">d</font>,<font color="Maroon" title="c22">d1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d2</font> &amp; <font color="Maroon" title="c23">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">G12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">d1</font> &amp; <font color="Maroon" title="c24">G12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">d</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E12:128_4"/><span class="lab"><font color="Green" title="E85">A20</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c21">d</font>,<font color="Maroon" title="c22">d1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">c11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">c21</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128_4"><span class="lab"><font color="Green" title="E84">A19</font></span></a>, <a class="txt" href="#E3:128_4"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="txt" href="#E5:128_4"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="txt" href="#E6:128_4"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>;<br/>
<span class="lab"><font color="Green" title="E86">A21</font></span>: <font color="Maroon" title="c21">d</font> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">C1</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c21">d</font>,<font color="Maroon" title="c22">d1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
<br/>.= 
<font color="Maroon" title="c8">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_4"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
;<br/>
<span class="lab"><font color="Green" title="E87">A22</font></span>: <font color="Maroon" title="c22">d1</font> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">C1</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c21">d</font>,<font color="Maroon" title="c22">d1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
<br/>.= 
<font color="Maroon" title="c9">c11</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">c21</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_4"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E15:128_4"/>
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_4"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="txt" href="#E5:128_4"><span class="lab"><font color="Green" title="E81">A16</font></span></a>, <a class="txt" href="#E8:128_4"><span class="lab"><font color="Green" title="E83">A18</font></span></a>, <a class="txt" href="#E6:128_4"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="txt" href="#E13:128_4"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="txt" href="#E11:128_4"><span class="lab"><font color="Green" title="E84">A19</font></span></a>, <a class="txt" href="#E14:128_4"><span class="lab"><font color="Green" title="E87">A22</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="E10:128"/><span class="lab"><font color="Green" title="E79">A23</font></span>: 
<font color="Maroon" title="c5">E</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="#E8:128"><span class="lab"><font color="Green" title="E78">A13</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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">  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="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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> ) )</span><br/>

<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="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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="c2">C1</font></span>)</span>,<font color="Maroon" title="c3">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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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="E11:128"/><span class="lab"><font color="Green" title="E80">A24</font></span>: 
<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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_5"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:128_5"/>
( <font color="Maroon" title="c6">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c6">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_5_1"><span class="kw">suppose </span></a><a NAME="E1:128_5_1"/>
<font color="Maroon" title="c6">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">D0</font> = <font color="Maroon" title="c6">D</font> as   <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> ;<br/><span class="kw">set </span><font color="Maroon" title="c9">H</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c8">D0</font>,<font color="Maroon" title="c1">C</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">H</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c8">D0</font>,<font color="Maroon" title="c1">C</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c1">C</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">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="c10">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="c10">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="c2">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>,<font color="Maroon" title="c1">C</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">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="c2">C1</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c10">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="E4:128_5_1"/>
( <font color="Maroon" title="c10">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="c10">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="c2">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>,<font color="Maroon" title="c1">C</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">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="c2">C1</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c10">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_5_2"><span class="kw">suppose </span></a><a NAME="E1:128_5_2"/><span class="lab"><font color="Green" title="E81">A25</font></span>: 
 not <font color="Maroon" title="c6">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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>,<font color="Maroon" title="c1">C</font> 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="c2">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>,<font color="Maroon" title="c1">C</font>  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="c2">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/><div class="add"><a NAME="E2:128_5_2"/><span class="lab"><font color="Green" title="E82">A26</font></span>: 
 for <font color="Olive" title="b1">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>  ex <font color="Olive" title="b2">F1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( (  for <font color="Olive" title="b3">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Olive" title="b2">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">d</font>,<font color="Olive" title="b3">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) &amp; ( <font color="Olive" title="b1">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Olive" title="b2">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  ) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">d</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <br/>( (  for <font color="Olive" title="b2">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Olive" title="b1">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Olive" title="b2">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) &amp; ( <font color="Maroon" title="c8">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> );<br/>
<a NAME="E1:128_5_2_1"/><span class="lab"><font color="Green" title="E83">A27</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">c1</font> = <font color="Maroon" title="c9">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</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#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>] )</span><br/>

<a NAME="E3:128_5_2_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c3">C2</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:128_5_2_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</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">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>]</span><br/>

<span class="kw">thus </span><a NAME="E5:128_5_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c10">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</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">consider </span><font color="Maroon" title="c9">F1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:128_5_2_1"/><span class="lab"><font color="Green" title="E84">A28</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:128_5_2_1"><span class="lab"><font color="Green" title="E83">A27</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c9">F1</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">F1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) &amp; ( <font color="Maroon" title="c8">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c10">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:128_5_2_1"/><span class="lab"><font color="Green" title="E85">A29</font></span>: 
 for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</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">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c10">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">c1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/>

<a NAME="E1:128_5_2_1_2"/>
<font color="Maroon" title="c11">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">C1</font>
 
;<br/>
<a NAME="E2:128_5_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A30</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">c1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E5:128_5_2_1_2"/><span class="lab"><font color="Green" title="E87">A31</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c2</font> &amp; <font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c13">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_1"><span class="lab"><font color="Green" title="E84">A28</font></span></a>, <a class="txt" href="#E2:128_5_2_1_2"><span class="lab"><font color="Green" title="E86">A30</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:128_5_2_1_2"/>
<font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_1_2"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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">thus </span><a NAME="E6:128_5_2_1"/>
( <font color="Maroon" title="c8">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c10">F1</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:128_5_2_1_3"/><span class="lab"><font color="Green" title="E86">A32</font></span>: 
<font color="Maroon" title="c8">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c10">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> </span><br/>

<a NAME="E2:128_5_2_1_3"/>
 for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>  st <font color="Olive" title="b1">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">c1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c11">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_1_3_1"/>
<font color="Maroon" title="c11">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<a NAME="E2:128_5_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A33</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E1:128_5_2_1_3"><span class="lab"><font color="Green" title="E86">A32</font></span></a>, <a class="ref" href="cat_8.html#T56" target="_self" onmouseover="rs('cat_8/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E3:128_5_2_1_3_1"/>
<font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_1"><span class="lab"><font color="Green" title="E85">A29</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:128_5_2_1_3_1"/>
<font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_1_3_1"><span class="lab"><font color="Green" title="E87">A33</font></span></a>, <a class="txt" href="#E11:128"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</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>
<a NAME="E3:128_5_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A34</font></span>: 
<font color="Maroon" title="c10">F1</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/>
<a NAME="E4:128_5_2_1_3"/>
 for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>  st <font color="Olive" title="b1">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">c2</font> holds <br/>( <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">c2</font> &amp; <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">c2</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">c1</font>, <font color="Maroon" title="c12">c2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c11">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">c2</font> implies ( <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font> &amp; <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_1_3_2"/><span class="lab"><font color="Green" title="E88">A35</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">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="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font> &amp; <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font></span>)</span> )</span><br/>

<a NAME="E2:128_5_2_1_3_2"/><span class="lab"><font color="Green" title="E89">A36</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E1:128_5_2_1_3"><span class="lab"><font color="Green" title="E86">A32</font></span></a>, <a class="ref" href="cat_6.html#T24" onmouseover="rs('cat_6/T24')" onmouseout="rh()">CAT_6:24</a></span>;<br/>
<a NAME="E3:128_5_2_1_3_2"/><span class="lab"><font color="Green" title="E90">A37</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E88">A35</font></span></a>, <a class="txt" href="#E2:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E89">A36</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<a NAME="E4:128_5_2_1_3_2"/><span class="lab"><font color="Green" title="E91">A38</font></span>: 
<font color="Maroon" title="c7">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E5:128_5_2_1_3_2"/><span class="lab"><font color="Green" title="E92">A39</font></span>: 
( <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_1"><span class="lab"><font color="Green" title="E85">A29</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:128_5_2_1_3_2"/>
<font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E91">A38</font></span></a>, <a class="txt" href="#E3:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E90">A37</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span> = 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_1"><span class="lab"><font color="Green" title="E85">A29</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c8">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">c2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E89">A36</font></span></a>, <a class="txt" href="#E1:128_5_2_1_3"><span class="lab"><font color="Green" title="E86">A32</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E88">A35</font></span></a>, <a class="txt" href="#E2:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E89">A36</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">c2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E91">A38</font></span></a>, <a class="txt" href="#E5:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E92">A39</font></span></a>, <a class="txt" href="#E3:128_5_2_1_3_2"><span class="lab"><font color="Green" title="E90">A37</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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="E5:128_5_2_1_3"/>
<font color="Maroon" title="c10">F1</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="#E3:128_5_2_1_3"><span class="lab"><font color="Green" title="E87">A34</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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>


</div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font> ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b6">T</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b3">d2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">d1</font> &amp; <font color="Olive" title="b2">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">d2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b5">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b5">F2</font> &amp; (  for <font color="Olive" title="b7">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ( <font color="Olive" title="b4">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b2">d1</font>,<font color="Olive" title="b7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Olive" title="b5">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b3">d2</font>,<font color="Olive" title="b7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Olive" title="b6">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">d</font>,<font color="Olive" title="b7">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) ) );<br/><a NAME="E3:128_5_2"/><span class="lab"><font color="Green" title="E83">A40</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c8">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">d2</font>, <font color="Maroon" title="c11">d1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font><span class="kw"> such that </span><br/><a NAME="E4:128_5_2_2"/><span class="lab"><font color="Green" title="E84">A41</font></span>: 
( <font color="Maroon" title="c10">d2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c11">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c10">d2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c9">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">d1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_8.html#T5" target="_self" onmouseover="rs('cat_8/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">F1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E6:128_5_2_2"/><span class="lab"><font color="Green" title="E85">A42</font></span>: 
( (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d1</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) &amp; ( <font color="Maroon" title="c11">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c12">F1</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="#E2:128_5_2"><span class="lab"><font color="Green" title="E82">A26</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E8:128_5_2_2"/><span class="lab"><font color="Green" title="E86">A43</font></span>: 
( (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d2</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) &amp; ( <font color="Maroon" title="c10">d2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c13">F2</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="#E2:128_5_2"><span class="lab"><font color="Green" title="E82">A26</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">T</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E10:128_5_2_2"/><span class="lab"><font color="Green" title="E87">A44</font></span>: 
( (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) &amp; ( <font color="Maroon" title="c9">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c14">T</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="#E2:128_5_2"><span class="lab"><font color="Green" title="E82">A26</font></span></a></span>;<br/>
<a NAME="E11:128_5_2_2"/>
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>  st <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c16">f1</font>, <font color="Maroon" title="c17">f2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c16">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c17">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">f2</font> implies ( <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_2_1"/><span class="lab"><font color="Green" title="E88">A45</font></span>: 
( <font color="Maroon" title="c16">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c17">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f</font> or  not <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">f2</font> or ( <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:128_5_2_2_1"/><span class="lab"><font color="Green" title="E89">A46</font></span>: 
( <font color="Maroon" title="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">f2</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="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span> )</span><br/>

<a NAME="E3:128_5_2_2_1"/><span class="lab"><font color="Green" title="E90">A47</font></span>: 
( <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c16">f1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c17">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:128_5_2_2"><span class="lab"><font color="Green" title="E87">A44</font></span></a></span>;<br/>
<a NAME="E4:128_5_2_2_1"/><span class="lab"><font color="Green" title="E91">A48</font></span>: 
<font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d1</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128_5_2_2"><span class="lab"><font color="Green" title="E85">A42</font></span></a></span>;<br/>
<a NAME="E5:128_5_2_2_1"/><span class="lab"><font color="Green" title="E92">A49</font></span>: 
<font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d2</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:128_5_2_2"><span class="lab"><font color="Green" title="E86">A43</font></span></a></span>;<br/>
<a NAME="E6:128_5_2_2_1"/><span class="lab"><font color="Green" title="E93">A50</font></span>: 
<font color="Maroon" title="c7">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E7:128_5_2_2_1"/><span class="lab"><font color="Green" title="E94">A51</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c16">f1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d1</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_2_1"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:128_5_2_2_1"/>
<font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_2_1"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="txt" href="#E4:128_5_2_2_1"><span class="lab"><font color="Green" title="E91">A48</font></span></a>, <a class="txt" href="#E7:128_5_2_2_1"><span class="lab"><font color="Green" title="E94">A51</font></span></a>, <a class="txt" href="#E6:128_5_2_2_1"><span class="lab"><font color="Green" title="E93">A50</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span> )</span><br/>

<a NAME="E9:128_5_2_2_1"/><span class="lab"><font color="Green" title="E95">A52</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d2</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c17">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_2_1"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:128_5_2_2_1"/>
<font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_2_1"><span class="lab"><font color="Green" title="E92">A49</font></span></a>, <a class="txt" href="#E3:128_5_2_2_1"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="txt" href="#E6:128_5_2_2_1"><span class="lab"><font color="Green" title="E93">A50</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> &amp; <font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> = 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:128_5_2_2"><span class="lab"><font color="Green" title="E87">A44</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_2_1"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E1:128_5_2_2_1"><span class="lab"><font color="Green" title="E88">A45</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">d1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c16">f1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d1</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_2_1"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c16">f1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d1</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_2_1"><span class="lab"><font color="Green" title="E94">A51</font></span></a>, <a class="txt" href="#E6:128_5_2_2_1"><span class="lab"><font color="Green" title="E93">A50</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_2_1"><span class="lab"><font color="Green" title="E91">A48</font></span></a>, <a class="txt" href="#E10:128_5_2_2"><span class="lab"><font color="Green" title="E87">A44</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="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> = 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:128_5_2_2"><span class="lab"><font color="Green" title="E87">A44</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">f2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_2_1"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E1:128_5_2_2_1"><span class="lab"><font color="Green" title="E88">A45</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">d2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">f2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d2</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c17">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_5_2_2_1"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d2</font>,<font color="Maroon" title="c15">f</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c17">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_2_1"><span class="lab"><font color="Green" title="E95">A52</font></span></a>, <a class="txt" href="#E6:128_5_2_2_1"><span class="lab"><font color="Green" title="E93">A50</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_2_1"><span class="lab"><font color="Green" title="E92">A49</font></span></a>, <a class="txt" href="#E10:128_5_2_2"><span class="lab"><font color="Green" title="E87">A44</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>
<a NAME="E12:128_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A53</font></span>: 
<font color="Maroon" title="c14">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a>, <a class="txt" href="#E6:128_5_2_2"><span class="lab"><font color="Green" title="E85">A42</font></span></a>, <a class="txt" href="#E8:128_5_2_2"><span class="lab"><font color="Green" title="E86">A43</font></span></a>, <a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a></span>;<br/>
<a NAME="E13:128_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A54</font></span>: 
<font color="Maroon" title="c12">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c13">F2</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">T</font> = <font color="Maroon" title="c14">T</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2_2"><span class="lab"><font color="Green" title="E88">A53</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>] )</span><br/>

<a NAME="E15:128_5_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:128_5_2_2"><span class="lab"><font color="Green" title="E89">A54</font></span></a>, <a class="txt" href="#E6:128_5_2_2"><span class="lab"><font color="Green" title="E85">A42</font></span></a>, <a class="txt" href="#E8:128_5_2_2"><span class="lab"><font color="Green" title="E86">A43</font></span></a>, <a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:128_5_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#D28" target="_self" onmouseover="rs('cat_8/D28')" onmouseout="rh()">Def28</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">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]</span><br/>

<span class="kw">thus </span><a NAME="E17:128_5_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_2"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="txt" href="#E6:128_5_2_2"><span class="lab"><font color="Green" title="E85">A42</font></span></a>, <a class="txt" href="#E8:128_5_2_2"><span class="lab"><font color="Green" title="E86">A43</font></span></a>, <a class="txt" href="#E10:128_5_2_2"><span class="lab"><font color="Green" title="E87">A44</font></span></a>, <a class="txt" href="#E13:128_5_2_2"><span class="lab"><font color="Green" title="E89">A54</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">consider </span><font color="Maroon" title="c8">H</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E5:128_5_2"/><span class="lab"><font color="Green" title="E84">A55</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:128_5_2"><span class="lab"><font color="Green" title="E83">A40</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c9">H</font> = <font color="Maroon" title="c8">H</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c1">C</font> ;<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="c2">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>,<font color="Maroon" title="c1">C</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">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="c2">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/><a NAME="E7:128_5_2"/><span class="lab"><font color="Green" title="E85">A56</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>  st <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_3"/><span class="lab"><font color="Green" title="E86">A57</font></span>: 
<font color="Maroon" title="c10">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">f</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E3:128_5_2_3"/>
 not  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a></span>;<br/>
<a NAME="E4:128_5_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font>
 
;<br/>
<a NAME="E5:128_5_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c13">d1</font>, <font color="Maroon" title="c14">d2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>, <font color="Maroon" title="c15">F1</font>, <font color="Maroon" title="c16">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c17">T</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c16">F2</font><span class="kw"> such that </span><br/><a NAME="E7:128_5_2_3"/><span class="lab"><font color="Green" title="E87">A58</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c14">d2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c12">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">d1</font> &amp; <font color="Maroon" title="c13">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c14">d2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c16">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c15">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F2</font> &amp; (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ( <font color="Maroon" title="c15">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c13">d1</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c16">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d2</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c17">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2"><span class="lab"><font color="Green" title="E84">A55</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E88">A59</font></span>: <font color="Maroon" title="c14">d2</font> = 
<font color="Maroon" title="c14">d2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="txt" href="#E1:128_5_2_3"><span class="lab"><font color="Green" title="E86">A57</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c12">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<span class="lab"><font color="Green" title="E89">A60</font></span>: <font color="Maroon" title="c13">d1</font> = 
<font color="Maroon" title="c12">d</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="txt" href="#E1:128_5_2_3"><span class="lab"><font color="Green" title="E86">A57</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c12">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<a NAME="E10:128_5_2_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon" title="c15">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies <font color="Maroon" title="c15">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_3_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c15">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">c1</font> = <font color="Maroon" title="c18">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c15">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> = 
<font color="Maroon" title="c15">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c13">d1</font>,<font color="Maroon" title="c19">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="txt" href="#E8:128_5_2_3"><span class="lab"><font color="Green" title="E88">A59</font></span></a>, <a class="txt" href="#E9:128_5_2_3"><span class="lab"><font color="Green" title="E89">A60</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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>
<a NAME="E11:128_5_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A61</font></span>: 
<font color="Maroon" title="c15">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<a NAME="E12:128_5_2_3"/><span class="lab"><font color="Green" title="E91">A62</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies <font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_3_4"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">c1</font> = <font color="Maroon" title="c18">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c16">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> = 
<font color="Maroon" title="c16">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d2</font>,<font color="Maroon" title="c19">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c19">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="txt" href="#E8:128_5_2_3"><span class="lab"><font color="Green" title="E88">A59</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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>
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font> = 
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c15">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="txt" href="#E12:128_5_2_3"><span class="lab"><font color="Green" title="E91">A62</font></span></a>, <a class="txt" href="#E11:128_5_2_3"><span class="lab"><font color="Green" title="E90">A61</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:128_5_2_3"/>
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E7:128_5_2_3"><span class="lab"><font color="Green" title="E87">A58</font></span></a>, <a class="ref" href="cat_8.html#T64" target="_self" onmouseover="rs('cat_8/T64')" onmouseout="rh()">Th64</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><a NAME="E8:128_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A63</font></span>: 
<font color="Maroon" title="c9">H</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/><a NAME="E9:128_5_2"/>
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>  st <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> holds <br/>( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f1</font>, <font color="Maroon" title="c11">f2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">f2</font> implies ( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font> &amp; <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_4"/><span class="lab"><font color="Green" title="E87">A64</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">f2</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> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font> &amp; <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">f1</font>, <font color="Maroon" title="c13">x2</font> = <font color="Maroon" title="c11">f2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E3:128_5_2_4"/><span class="lab"><font color="Green" title="E88">A65</font></span>: 
 not  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a></span>;<br/>
<a NAME="E4:128_5_2_4"/><span class="kw">then </span>
( <font color="Maroon" title="c10">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font> &amp; <font color="Maroon" title="c11">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font> )
 
;<br/>
<a NAME="E5:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A66</font></span>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">d1</font>, <font color="Maroon" title="c15">d11</font>, <font color="Maroon" title="c16">d12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>, <font color="Maroon" title="c17">F11</font>, <font color="Maroon" title="c18">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c19">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c17">F11</font>,<font color="Maroon" title="c18">F12</font><span class="kw"> such that </span><br/><a NAME="E7:128_5_2_4"/><span class="lab"><font color="Green" title="E90">A67</font></span>: 
( <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">d1</font> &amp; <font color="Maroon" title="c16">d12</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">d1</font> &amp; <font color="Maroon" title="c14">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">d11</font> &amp; <font color="Maroon" title="c15">d11</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c16">d12</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F11</font>,<font color="Maroon" title="c18">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c18">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c17">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c18">F12</font> &amp; (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ( <font color="Maroon" title="c17">F11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c15">d11</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c18">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c19">T1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_4"><span class="lab"><font color="Green" title="E89">A66</font></span></a>, <a class="txt" href="#E5:128_5_2"><span class="lab"><font color="Green" title="E84">A55</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">d2</font>, <font color="Maroon" title="c21">d21</font>, <font color="Maroon" title="c22">d22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>, <font color="Maroon" title="c23">F21</font>, <font color="Maroon" title="c24">F22</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c25">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c23">F21</font>,<font color="Maroon" title="c24">F22</font><span class="kw"> such that </span><br/><a NAME="E9:128_5_2_4"/><span class="lab"><font color="Green" title="E91">A68</font></span>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">d2</font> &amp; <font color="Maroon" title="c22">d22</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">d2</font> &amp; <font color="Maroon" title="c20">d2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c21">d21</font> &amp; <font color="Maroon" title="c21">d21</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c22">d22</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F21</font>,<font color="Maroon" title="c24">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c25">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">F21</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F22</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F21</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F22</font> &amp; (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ( <font color="Maroon" title="c23">F21</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c21">d21</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c24">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c22">d22</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c25">T2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_4"><span class="lab"><font color="Green" title="E89">A66</font></span></a>, <a class="txt" href="#E5:128_5_2"><span class="lab"><font color="Green" title="E84">A55</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">x12</font> = <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E11:128_5_2_4"/>
<font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_4"><span class="lab"><font color="Green" title="E88">A65</font></span></a></span>;<br/>
<a NAME="E12:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A69</font></span>: 
<font color="Maroon" title="c26">x12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">d3</font>, <font color="Maroon" title="c28">d31</font>, <font color="Maroon" title="c29">d32</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>, <font color="Maroon" title="c30">F31</font>, <font color="Maroon" title="c31">F32</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c32">T3</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c30">F31</font>,<font color="Maroon" title="c31">F32</font><span class="kw"> such that </span><br/><a NAME="E14:128_5_2_4"/><span class="lab"><font color="Green" title="E93">A70</font></span>: 
( <font color="Maroon" title="c26">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">d3</font> &amp; <font color="Maroon" title="c29">d32</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">d3</font> &amp; <font color="Maroon" title="c27">d3</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c28">d31</font> &amp; <font color="Maroon" title="c28">d31</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c29">d32</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">F31</font>,<font color="Maroon" title="c31">F32</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c32">T3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c30">F31</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c31">F32</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c30">F31</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c31">F32</font> &amp; (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ( <font color="Maroon" title="c30">F31</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c28">d31</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c31">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c29">d32</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c32">T3</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c27">d3</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2_4"><span class="lab"><font color="Green" title="E92">A69</font></span></a>, <a class="txt" href="#E5:128_5_2"><span class="lab"><font color="Green" title="E84">A55</font></span></a></span>;<br/>
<a NAME="E15:128_5_2_4"/><span class="lab"><font color="Green" title="E94">A71</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c20">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c14">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_7.html#T5" onmouseover="rs('cat_7/T5')" onmouseout="rh()">CAT_7:5</a>, <a class="txt" href="#E1:128_5_2_4"><span class="lab"><font color="Green" title="E87">A64</font></span></a>, <a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E95">A72</font></span>: <font color="Maroon" title="c16">d12</font> = 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c14">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="ref" href="cat_6.html#T27" onmouseover="rs('cat_6/T27')" onmouseout="rh()">CAT_6:27</a></span>
<br/>.= 
<font color="Maroon" title="c21">d21</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:128_5_2_4"><span class="lab"><font color="Green" title="E94">A71</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="ref" href="cat_6.html#T26" onmouseover="rs('cat_6/T26')" onmouseout="rh()">CAT_6:26</a></span>
;<br/>
<a NAME="E17:128_5_2_4"/><span class="lab"><font color="Green" title="E96">A73</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon" title="c18">F12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">F21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies <font color="Maroon" title="c18">F12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">F21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_4_2"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c18">F12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">F21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">c1</font> = <font color="Maroon" title="c33">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c18">F12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> = 
<font color="Maroon" title="c18">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c34">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Maroon" title="c34">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c23">F21</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c34">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_5_2_4"><span class="lab"><font color="Green" title="E95">A72</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c23">F21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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>
<a NAME="E18:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A74</font></span>: 
<font color="Maroon" title="c18">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">F21</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">T2</font> = <font color="Maroon" title="c25">T2</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c18">F12</font>,<font color="Maroon" title="c24">F22</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:128_5_2_4"><span class="lab"><font color="Green" title="E96">A73</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<span class="lab"><font color="Green" title="E98">A75</font></span>: <font color="Maroon" title="c28">d31</font> = 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a>, <a class="ref" href="cat_6.html#T26" onmouseover="rs('cat_6/T26')" onmouseout="rh()">CAT_6:26</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c14">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E1:128_5_2_4"><span class="lab"><font color="Green" title="E87">A64</font></span></a>, <a class="ref" href="cat_7.html#T4" onmouseover="rs('cat_7/T4')" onmouseout="rh()">CAT_7:4</a></span>
<br/>.= 
<font color="Maroon" title="c15">d11</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="ref" href="cat_6.html#T26" onmouseover="rs('cat_6/T26')" onmouseout="rh()">CAT_6:26</a></span>
;<br/>
<a NAME="E21:128_5_2_4"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon" title="c30">F31</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">F11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_4_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies <font color="Maroon" title="c30">F31</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">F11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_4_4"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c30">F31</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">F11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">c1</font> = <font color="Maroon" title="c34">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c30">F31</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> = 
<font color="Maroon" title="c30">F31</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c28">d31</font>,<font color="Maroon" title="c35">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">F11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:128_5_2_4"><span class="lab"><font color="Green" title="E98">A75</font></span></a>, <a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">F11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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>
<a NAME="E22:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A76</font></span>: 
<font color="Maroon" title="c30">F31</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">F11</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<span class="lab"><font color="Green" title="E100">A77</font></span>: <font color="Maroon" title="c29">d32</font> = 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a>, <a class="ref" href="cat_6.html#T27" onmouseover="rs('cat_6/T27')" onmouseout="rh()">CAT_6:27</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c20">d2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="txt" href="#E1:128_5_2_4"><span class="lab"><font color="Green" title="E87">A64</font></span></a>, <a class="ref" href="cat_7.html#T4" onmouseover="rs('cat_7/T4')" onmouseout="rh()">CAT_7:4</a></span>
<br/>.= 
<font color="Maroon" title="c22">d22</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="ref" href="cat_6.html#T27" onmouseover="rs('cat_6/T27')" onmouseout="rh()">CAT_6:27</a></span>
;<br/>
<a NAME="E24:128_5_2_4"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon" title="c31">F32</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">F22</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_4_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies <font color="Maroon" title="c31">F32</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">F22</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_4_6"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c31">F32</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">F22</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">c1</font> = <font color="Maroon" title="c34">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c31">F32</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> = 
<font color="Maroon" title="c31">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c29">d32</font>,<font color="Maroon" title="c35">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c24">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:128_5_2_4"><span class="lab"><font color="Green" title="E100">A77</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c24">F22</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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>
<a NAME="E25:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A78</font></span>: 
<font color="Maroon" title="c31">F32</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c24">F22</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<a NAME="E26:128_5_2_4"/><span class="lab"><font color="Green" title="E102">A79</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> holds <br/><font color="Maroon" title="c32">T3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_4_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> implies <font color="Maroon" title="c32">T3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_4_7"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c32">T3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">c</font> = <font color="Maroon" title="c34">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c36">c2</font>, <font color="Maroon" title="c37">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E4:128_5_2_4_7"/><span class="lab"><font color="Green" title="E103">A80</font></span>: 
( <font color="Maroon" title="c36">c2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c37">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c36">c2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c35">c</font> &amp; <font color="Maroon" title="c35">c</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c37">c1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T5" target="_self" onmouseover="rs('cat_8/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E5:128_5_2_4_7"/><span class="lab"><font color="Green" title="E104">A81</font></span>: 
<font color="Maroon" title="c7">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E6:128_5_2_4_7"/><span class="lab"><font color="Green" title="E105">A82</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c36">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="txt" href="#E16:128_5_2_4"><span class="lab"><font color="Green" title="E95">A72</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E106">A83</font></span>: <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c36">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">d2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">d12</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c36">c2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c35">c</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_5_2_4"><span class="lab"><font color="Green" title="E95">A72</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c36">c2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c35">c</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_5_2_4"><span class="lab"><font color="Green" title="E95">A72</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<a NAME="E8:128_5_2_4_7"/><span class="lab"><font color="Green" title="E107">A84</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c37">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="txt" href="#E1:128_5_2_4"><span class="lab"><font color="Green" title="E87">A64</font></span></a>, <a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="lab"><font color="Green" title="E108">A85</font></span>: <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c37">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">d2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">d1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">c</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c37">c1</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2_4"><span class="lab"><font color="Green" title="E87">A64</font></span></a>, <a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c27">d3</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a>, <a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c32">T3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> = 
<font color="Maroon" title="c32">T3</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c37">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_4_7"><span class="lab"><font color="Green" title="E108">A85</font></span></a>, <a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c36">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c37">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4_7"><span class="lab"><font color="Green" title="E106">A83</font></span></a>, <a class="txt" href="#E5:128_5_2_4_7"><span class="lab"><font color="Green" title="E104">A81</font></span></a>, <a class="txt" href="#E8:128_5_2_4_7"><span class="lab"><font color="Green" title="E107">A84</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c36">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c37">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128_5_2_4_7"><span class="lab"><font color="Green" title="E105">A82</font></span></a>, <a class="txt" href="#E5:128_5_2_4_7"><span class="lab"><font color="Green" title="E104">A81</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c36">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">d12</font>,<font color="Maroon" title="c35">c</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">T1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d2</font>,<font color="Maroon" title="c36">c2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">T1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">c2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">T1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:128_5_2_4"><span class="lab"><font color="Green" title="E97">A74</font></span></a>, <a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="txt" href="#E4:128_5_2_4_7"><span class="lab"><font color="Green" title="E103">A80</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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="lab"><font color="Green" title="E103">A86</font></span>: <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font> = 
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F12</font>,<font color="Maroon" title="c24">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c33">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:128_5_2_4"><span class="lab"><font color="Green" title="E96">A73</font></span></a>, <a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>
;<br/>
<a NAME="E28:128_5_2_4"/><span class="lab"><font color="Green" title="E104">A87</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F11</font>,<font color="Maroon" title="c18">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<span class="lab"><font color="Green" title="E105">A88</font></span>: <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">F31</font>,<font color="Maroon" title="c31">F32</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c32">T3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F11</font>,<font color="Maroon" title="c24">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:128_5_2_4"><span class="lab"><font color="Green" title="E99">A76</font></span></a>, <a class="txt" href="#E25:128_5_2_4"><span class="lab"><font color="Green" title="E101">A78</font></span></a>, <a class="txt" href="#E26:128_5_2_4"><span class="lab"><font color="Green" title="E102">A79</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>
;<br/>
<a NAME="E30:128_5_2_4"/><span class="lab"><font color="Green" title="E106">A89</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Olive" title="b3">t</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#D28" target="_self" onmouseover="rs('cat_8/D28')" onmouseout="rh()">Def28</a></span>;<br/>
<a NAME="E31:128_5_2_4"/>
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_4"><span class="lab"><font color="Green" title="E91">A68</font></span></a>, <a class="txt" href="#E30:128_5_2_4"><span class="lab"><font color="Green" title="E106">A89</font></span></a></span>;<br/>
<a NAME="E32:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A90</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E33:128_5_2_4"/>
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:128_5_2_4"><span class="lab"><font color="Green" title="E90">A67</font></span></a>, <a class="txt" href="#E30:128_5_2_4"><span class="lab"><font color="Green" title="E106">A89</font></span></a></span>;<br/>
<a NAME="E34:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A91</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E35:128_5_2_4"/>
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_4"><span class="lab"><font color="Green" title="E93">A70</font></span></a>, <a class="txt" href="#E30:128_5_2_4"><span class="lab"><font color="Green" title="E106">A89</font></span></a></span>;<br/>
<a NAME="E36:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A92</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<a NAME="E37:128_5_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#NK13" title="CAT_8:NK.13">KuratowskiPair</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font> ex <font color="Olive" title="b7">t1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">t2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">t2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">t1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:128_5_2_4"><span class="lab"><font color="Green" title="E103">A86</font></span></a>, <a class="txt" href="#E28:128_5_2_4"><span class="lab"><font color="Green" title="E104">A87</font></span></a>, <a class="txt" href="#E29:128_5_2_4"><span class="lab"><font color="Green" title="E105">A88</font></span></a>, <a class="txt" href="#E32:128_5_2_4"><span class="lab"><font color="Green" title="E107">A90</font></span></a>, <a class="txt" href="#E34:128_5_2_4"><span class="lab"><font color="Green" title="E108">A91</font></span></a>, <a class="txt" href="#E36:128_5_2_4"><span class="lab"><font color="Green" title="E109">A92</font></span></a></span>;<br/>
<a NAME="E38:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A93</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#NK13" title="CAT_8:NK.13">KuratowskiPair</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#D28" target="_self" onmouseover="rs('cat_8/D28')" onmouseout="rh()">Def28</a></span>;<br/>
<a NAME="E39:128_5_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A94</font></span>: 
 <a href="cat_8.html#NK13" title="CAT_8:NK.13">KuratowskiPair</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:128_5_2_4"/>
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</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">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">f2</font></span>)</span> = 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_8.html#NK13" title="CAT_8:NK.13">KuratowskiPair</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:128_5_2_4"><span class="lab"><font color="Green" title="E111">A94</font></span></a>, <a class="txt" href="#E38:128_5_2_4"><span class="lab"><font color="Green" title="E110">A93</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>
<br/>.= 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:128_5_2_4"><span class="lab"><font color="Green" title="E111">A94</font></span></a>, <a class="ref" href="cat_6.html#D3" onmouseover="rs('cat_6/D3')" onmouseout="rh()">CAT_6:def 3</a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</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="E10:128_5_2"/><span class="lab"><font color="Green" title="E87">A95</font></span>: 
<font color="Maroon" title="c9">H</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="#E8:128_5_2"><span class="lab"><font color="Green" title="E86">A63</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</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="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="c2">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>,<font color="Maroon" title="c1">C</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">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="c2">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/><a NAME="E11:128_5_2"/><span class="lab"><font color="Green" title="E88">A96</font></span>: 
 for <font color="Olive" title="b1">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font><br/>  for <font color="Olive" title="b2">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">d</font></span>)</span>,<font color="Olive" title="b2">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">d</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font></span>)</span>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">c1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E2:128_5_2_5"/>
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>
 
;<br/>
<a NAME="E3:128_5_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">c</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c14">c11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c15">d1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C1</font></span>)</span>, <font color="Maroon" title="c16">c22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c17">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E5:128_5_2_5"/><span class="lab"><font color="Green" title="E89">A97</font></span>: 
( <font color="Maroon" title="c15">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c13">c</font>,<font color="Maroon" title="c14">c11</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d1</font> &amp; <font color="Maroon" title="c5">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c22</font> &amp; <font color="Maroon" title="c16">c22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">c11</font> &amp; <font color="Maroon" title="c17">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128"><span class="lab"><font color="Green" title="E76">A6</font></span></a></span>;<br/>
<a NAME="E6:128_5_2_5"/><span class="lab"><font color="Green" title="E90">A98</font></span>: 
( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_5"><span class="lab"><font color="Green" title="E89">A97</font></span></a>, <a class="ref" href="cat_8.html#T53" target="_self" onmouseover="rs('cat_8/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c10">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E8:128_5_2_5"/>
 not  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a></span>;<br/>
<a NAME="E9:128_5_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c6">D</font>
 
;<br/>
<a NAME="E10:128_5_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">d1</font>, <font color="Maroon" title="c20">d11</font>, <font color="Maroon" title="c21">d12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>, <font color="Maroon" title="c22">F1</font>, <font color="Maroon" title="c23">F2</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c24">T</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c22">F1</font>,<font color="Maroon" title="c23">F2</font><span class="kw"> such that </span><br/><a NAME="E12:128_5_2_5"/><span class="lab"><font color="Green" title="E91">A99</font></span>: 
( <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c21">d12</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c19">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">d11</font> &amp; <font color="Maroon" title="c20">d11</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c21">d12</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">F1</font>,<font color="Maroon" title="c23">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c24">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c22">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c23">F2</font> &amp; (  for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds <br/> ( <font color="Maroon" title="c22">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">d11</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c23">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c21">d12</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c24">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c19">d1</font>,<font color="Olive" title="b1">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2"><span class="lab"><font color="Green" title="E84">A55</font></span></a></span>;<br/>
<a NAME="E13:128_5_2_5"/><span class="lab"><font color="Green" title="E92">A100</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">F1</font>,<font color="Maroon" title="c23">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c24">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2_5"><span class="lab"><font color="Green" title="E91">A99</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<font color="Maroon" title="c5">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_5"><span class="lab"><font color="Green" title="E89">A97</font></span></a>, <a class="txt" href="#E6:128_5_2_5"><span class="lab"><font color="Green" title="E90">A98</font></span></a>, <a class="txt" href="#E13:128_5_2_5"><span class="lab"><font color="Green" title="E92">A100</font></span></a>, <a class="txt" href="#E12:128_5_2_5"><span class="lab"><font color="Green" title="E91">A99</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c11">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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><a NAME="E12:128_5_2"/><span class="lab"><font color="Green" title="E89">A101</font></span>: 
 for <font color="Olive" title="b1">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> holds  <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">c1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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"> <span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">c1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E2:128_5_2_6"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">C1</font>
 
;<br/>
<a NAME="E3:128_5_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A102</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">c1</font> = 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c10">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_6"><span class="lab"><font color="Green" title="E90">A102</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</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><a NAME="E13:128_5_2"/><span class="lab"><font color="Green" title="E90">A103</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C1</font></span>)</span> holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C1</font></span>)</span> implies <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_7"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C1</font></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="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</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="c2">C1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E3:128_5_2_7"/><span class="lab"><font color="Green" title="E91">A104</font></span>: 
<font color="Maroon" title="c9">H</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="c2">C1</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="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>, <font color="Maroon" title="c13">c1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font><span class="kw"> such that </span><br/><a NAME="E5:128_5_2_7"/><span class="lab"><font color="Green" title="E92">A105</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T52" target="_self" onmouseover="rs('cat_8/T52')" onmouseout="rh()">Th52</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<font color="Maroon" title="c7">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_7"><span class="lab"><font color="Green" title="E92">A105</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c13">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<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="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">c1</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2"><span class="lab"><font color="Green" title="E89">A101</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_5_2_7"><span class="lab"><font color="Green" title="E92">A105</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_8.html#T57" target="_self" onmouseover="rs('cat_8/T57')" onmouseout="rh()">Th57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:128"><span class="lab"><font color="Green" title="E79">A23</font></span></a>, <a class="txt" href="#E3:128_5_2_7"><span class="lab"><font color="Green" title="E91">A104</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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="E14:128_5_2"/>
<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="c2">C1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</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">  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>,<font color="Maroon" title="c1">C</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">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="c2">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>,<font color="Maroon" title="c1">C</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="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="c2">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="E15:128_5_2"/><span class="lab"><font color="Green" title="E91">A106</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="c2">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">assume </span><a NAME="E16:128_5_2"/><span class="lab"><font color="Green" title="E92">A107</font></span>: 
<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="c2">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/><a NAME="E17:128_5_2"/><span class="lab"><font color="Green" title="E93">A108</font></span>: 
 for <font color="Olive" title="b1">d</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>  st <font color="Olive" title="b1">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">d</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">d</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_8"/><span class="lab"><font color="Green" title="E94">A109</font></span>: 
<font color="Maroon" title="c11">d</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">F1</font> being   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E3:128_5_2_8"/><span class="lab"><font color="Green" title="E95">A110</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c12">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#T64" target="_self" onmouseover="rs('cat_8/T64')" onmouseout="rh()">Th64</a>, <a class="txt" href="#E7:128_5_2"><span class="lab"><font color="Green" title="E85">A56</font></span></a></span>;<br/>
<a NAME="E4:128_5_2_8"/>
<font color="Maroon" title="c10">H1</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">F2</font> being   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E6:128_5_2_8"/><span class="lab"><font color="Green" title="E96">A111</font></span>: 
<font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F2</font>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#T64" target="_self" onmouseover="rs('cat_8/T64')" onmouseout="rh()">Th64</a>, <a class="txt" href="#E1:128_5_2_8"><span class="lab"><font color="Green" title="E94">A109</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/>
<a NAME="E7:128_5_2_8"/>
<font color="Maroon" title="c12">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">F2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:128_5_2_8_1"/>
<font color="Maroon" title="c12">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">F2</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:128_5_2_8_1"/><span class="lab"><font color="Green" title="E97">A112</font></span>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">c1</font> = <font color="Maroon" title="c14">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_8_1"><span class="lab"><font color="Green" title="E97">A112</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E5:128_5_2_8_1"/><span class="lab"><font color="Green" title="E98">A113</font></span>: 
<font color="Maroon" title="c9">H</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="c2">C1</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="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<span class="lab"><font color="Green" title="E99">A114</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128"><span class="lab"><font color="Green" title="E79">A23</font></span></a>, <a class="txt" href="#E5:128_5_2_8_1"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<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="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">c1</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_8.html#T57" target="_self" onmouseover="rs('cat_8/T57')" onmouseout="rh()">Th57</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2"><span class="lab"><font color="Green" title="E89">A101</font></span></a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">c01</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c17">c11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c18">c12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c19">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E8:128_5_2_8_1"/><span class="lab"><font color="Green" title="E100">A115</font></span>: 
( <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c16">c01</font>,<font color="Maroon" title="c17">c11</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c12</font> &amp; <font color="Maroon" title="c18">c12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c17">c11</font> &amp; <font color="Maroon" title="c19">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c01</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E9:128_5_2_8_1"/><span class="lab"><font color="Green" title="E101">A116</font></span>: 
( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c01</font> &amp; <font color="Maroon" title="c15">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:128_5_2_8_1"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="ref" href="cat_8.html#T53" target="_self" onmouseover="rs('cat_8/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E10:128_5_2_8_1"/><span class="lab"><font color="Green" title="E102">A117</font></span>: 
<font color="Maroon" title="c10">H1</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="c2">C1</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="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<span class="lab"><font color="Green" title="E103">A118</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128"><span class="lab"><font color="Green" title="E79">A23</font></span></a>, <a class="txt" href="#E10:128_5_2_8_1"><span class="lab"><font color="Green" title="E102">A117</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<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="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">c1</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_8.html#T57" target="_self" onmouseover="rs('cat_8/T57')" onmouseout="rh()">Th57</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2"><span class="lab"><font color="Green" title="E89">A101</font></span></a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">c02</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c21">c21</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c22">c22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c23">F22</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E13:128_5_2_8_1"/><span class="lab"><font color="Green" title="E104">A119</font></span>: 
( <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c20">c02</font>,<font color="Maroon" title="c21">c21</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font></span>)</span>,<font color="Maroon" title="c15">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">c22</font> &amp; <font color="Maroon" title="c22">c22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c21">c21</font> &amp; <font color="Maroon" title="c23">F22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c02</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E14:128_5_2_8_1"/><span class="lab"><font color="Green" title="E105">A120</font></span>: 
( <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c02</font> &amp; <font color="Maroon" title="c15">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c21</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:128_5_2_8_1"><span class="lab"><font color="Green" title="E104">A119</font></span></a>, <a class="ref" href="cat_8.html#T53" target="_self" onmouseover="rs('cat_8/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<font color="Maroon" title="c12">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c13">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_8"><span class="lab"><font color="Green" title="E95">A110</font></span></a>, <a class="txt" href="#E6:128_5_2_8"><span class="lab"><font color="Green" title="E96">A111</font></span></a>, <a class="txt" href="#E14:128_5_2_8_1"><span class="lab"><font color="Green" title="E105">A120</font></span></a>, <a class="txt" href="#E8:128_5_2_8_1"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="txt" href="#E9:128_5_2_8_1"><span class="lab"><font color="Green" title="E101">A116</font></span></a>, <a class="txt" href="#E13:128_5_2_8_1"><span class="lab"><font color="Green" title="E104">A119</font></span></a>, <a class="txt" href="#E6:128_5_2_8_1"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="txt" href="#E11:128_5_2_8_1"><span class="lab"><font color="Green" title="E103">A118</font></span></a>, <a class="txt" href="#E13:128_5_2"><span class="lab"><font color="Green" title="E90">A103</font></span></a>, <a class="txt" href="#E16:128_5_2"><span class="lab"><font color="Green" title="E92">A107</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>
<br/>.= 
<font color="Maroon" title="c13">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:128_5_2_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_8_1"><span class="lab"><font color="Green" title="E97">A112</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="E8:128_5_2_8"/>
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_8"><span class="lab"><font color="Green" title="E95">A110</font></span></a>, <a class="txt" href="#E6:128_5_2_8"><span class="lab"><font color="Green" title="E96">A111</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><a NAME="E18:128_5_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> holds <br/><font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font> implies <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:128_5_2_9"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">D</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c11">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">d1</font>, <font color="Maroon" title="c14">d2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D</font><span class="kw"> such that </span><br/><a NAME="E4:128_5_2_9"/><span class="lab"><font color="Green" title="E94">A121</font></span>: 
( <font color="Maroon" title="c13">d1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c14">d2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c13">d1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c12">d</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">d2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_8.html#T5" target="_self" onmouseover="rs('cat_8/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E5:128_5_2_9"/><span class="lab"><font color="Green" title="E95">A122</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">d2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">d1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_9"><span class="lab"><font color="Green" title="E94">A121</font></span></a>, <a class="ref" href="cat_6.html#T26" onmouseover="rs('cat_6/T26')" onmouseout="rh()">CAT_6:26</a>, <a class="ref" href="cat_6.html#T27" onmouseover="rs('cat_6/T27')" onmouseout="rh()">CAT_6:27</a></span>;<br/>
<a NAME="E6:128_5_2_9"/><span class="lab"><font color="Green" title="E96">A123</font></span>: 
( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">d1</font> &amp; <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">d2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_5_2_9"><span class="lab"><font color="Green" title="E94">A121</font></span></a>, <a class="txt" href="#E17:128_5_2"><span class="lab"><font color="Green" title="E93">A108</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E97">A124</font></span>:  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span> = 
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c12">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">d2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E5:128_5_2_9"><span class="lab"><font color="Green" title="E95">A122</font></span></a>, <a class="txt" href="#E6:128_5_2_9"><span class="lab"><font color="Green" title="E96">A123</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c10">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c12">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E5:128_5_2_9"><span class="lab"><font color="Green" title="E95">A122</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
;<br/>
<span class="lab"><font color="Green" title="E98">A125</font></span>:  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span> = 
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c12">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E5:128_5_2_9"><span class="lab"><font color="Green" title="E95">A122</font></span></a>, <a class="txt" href="#E6:128_5_2_9"><span class="lab"><font color="Green" title="E96">A123</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c10">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c12">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E5:128_5_2_9"><span class="lab"><font color="Green" title="E95">A122</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">F1</font>, <font color="Maroon" title="c16">F2</font> being   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c17">T</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c16">F2</font><span class="kw"> such that </span><br/><a NAME="E10:128_5_2_9"/><span class="lab"><font color="Green" title="E99">A126</font></span>: 
( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c15">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F2</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#T65" target="_self" onmouseover="rs('cat_8/T65')" onmouseout="rh()">Th65</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">F11</font>, <font color="Maroon" title="c19">F12</font> being   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c20">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c18">F11</font>,<font color="Maroon" title="c19">F12</font><span class="kw"> such that </span><br/><a NAME="E12:128_5_2_9"/><span class="lab"><font color="Green" title="E100">A127</font></span>: 
( <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F11</font>,<font color="Maroon" title="c19">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F11</font>,<font color="Maroon" title="c18">F11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">F11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F12</font>,<font color="Maroon" title="c19">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="cat_8.html#T65" target="_self" onmouseover="rs('cat_8/T65')" onmouseout="rh()">Th65</a></span>;<br/>
<a NAME="E13:128_5_2_9"/><span class="lab"><font color="Green" title="E101">A128</font></span>: 
<font color="Maroon" title="c15">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">F11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:128_5_2_9"><span class="lab"><font color="Green" title="E99">A126</font></span></a>, <a class="txt" href="#E12:128_5_2_9"><span class="lab"><font color="Green" title="E100">A127</font></span></a>, <a class="txt" href="#E7:128_5_2_9"><span class="lab"><font color="Green" title="E97">A124</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E14:128_5_2_9"/><span class="lab"><font color="Green" title="E102">A129</font></span>: 
<font color="Maroon" title="c16">F2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c19">F12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:128_5_2_9"><span class="lab"><font color="Green" title="E99">A126</font></span></a>, <a class="txt" href="#E12:128_5_2_9"><span class="lab"><font color="Green" title="E100">A127</font></span></a>, <a class="txt" href="#E8:128_5_2_9"><span class="lab"><font color="Green" title="E98">A125</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E15:128_5_2_9"/><span class="lab"><font color="Green" title="E103">A130</font></span>: 
<font color="Maroon" title="c17">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">T1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="128_5_2_9_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:128_5_2_9_3"/>
<font color="Maroon" title="c17">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">T1</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:128_5_2_9_3"/><span class="lab"><font color="Green" title="E104">A131</font></span>: 
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> &amp; <font color="Maroon" title="c17">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">c1</font> = <font color="Maroon" title="c21">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_9_3"><span class="lab"><font color="Green" title="E104">A131</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E5:128_5_2_9_3"/><span class="lab"><font color="Green" title="E105">A132</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:128_5_2"><span class="lab"><font color="Green" title="E90">A103</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a>, <a class="txt" href="#E16:128_5_2"><span class="lab"><font color="Green" title="E92">A107</font></span></a></span>;<br/>
<a NAME="E6:128_5_2_9_3"/><span class="lab"><font color="Green" title="E106">A133</font></span>: 
<font color="Maroon" title="c9">H</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="c2">C1</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="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<span class="lab"><font color="Green" title="E107">A134</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128"><span class="lab"><font color="Green" title="E79">A23</font></span></a>, <a class="txt" href="#E6:128_5_2_9_3"><span class="lab"><font color="Green" title="E106">A133</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<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="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">c1</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128_5_2"><span class="lab"><font color="Green" title="E87">A95</font></span></a>, <a class="ref" href="cat_8.html#T57" target="_self" onmouseover="rs('cat_8/T57')" onmouseout="rh()">Th57</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2"><span class="lab"><font color="Green" title="E89">A101</font></span></a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">c01</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c24">c11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c25">c12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c26">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E9:128_5_2_9_3"/><span class="lab"><font color="Green" title="E108">A135</font></span>: 
( <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c23">c01</font>,<font color="Maroon" title="c24">c11</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c12</font> &amp; <font color="Maroon" title="c25">c12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c24">c11</font> &amp; <font color="Maroon" title="c26">F12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">c01</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E10:128_5_2_9_3"/><span class="lab"><font color="Green" title="E109">A136</font></span>: 
( <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">c01</font> &amp; <font color="Maroon" title="c22">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">c11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_5_2_9_3"><span class="lab"><font color="Green" title="E108">A135</font></span></a>, <a class="ref" href="cat_8.html#T53" target="_self" onmouseover="rs('cat_8/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E11:128_5_2_9_3"/><span class="lab"><font color="Green" title="E110">A137</font></span>: 
<font color="Maroon" title="c10">H1</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="c2">C1</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="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="ref" href="cat_8.html#D22" target="_self" onmouseover="rs('cat_8/D22')" onmouseout="rh()">Def22</a></span>;<br/>
<span class="lab"><font color="Green" title="E111">A138</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> = 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="p3">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C1</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128"><span class="lab"><font color="Green" title="E79">A23</font></span></a>, <a class="txt" href="#E11:128_5_2_9_3"><span class="lab"><font color="Green" title="E110">A137</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<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="c2">C1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">c1</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:128_5_2"><span class="lab"><font color="Green" title="E91">A106</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_8.html#T57" target="_self" onmouseover="rs('cat_8/T57')" onmouseout="rh()">Th57</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:128_5_2"><span class="lab"><font color="Green" title="E89">A101</font></span></a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">c02</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c28">c21</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c29">c22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c30">F22</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E14:128_5_2_9_3"/><span class="lab"><font color="Green" title="E112">A139</font></span>: 
( <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c27">c02</font>,<font color="Maroon" title="c28">c21</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp; <font color="Maroon" title="c5">E</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font></span>)</span>,<font color="Maroon" title="c22">c1</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">c22</font> &amp; <font color="Maroon" title="c29">c22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c28">c21</font> &amp; <font color="Maroon" title="c30">F22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">c02</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/>
<a NAME="E15:128_5_2_9_3"/><span class="lab"><font color="Green" title="E113">A140</font></span>: 
( <font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">c02</font> &amp; <font color="Maroon" title="c22">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">c21</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:128_5_2_9_3"><span class="lab"><font color="Green" title="E112">A139</font></span></a>, <a class="ref" href="cat_8.html#T53" target="_self" onmouseover="rs('cat_8/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E16:128_5_2_9_3"/>
<font color="Maroon" title="c17">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c22">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:128_5_2_9_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128_5_2_9_3"><span class="lab"><font color="Green" title="E104">A131</font></span></a>, <a class="txt" href="#E10:128_5_2_9_3"><span class="lab"><font color="Green" title="E109">A136</font></span></a>, <a class="txt" href="#E9:128_5_2_9_3"><span class="lab"><font color="Green" title="E108">A135</font></span></a>, <a class="txt" href="#E10:128_5_2_9"><span class="lab"><font color="Green" title="E99">A126</font></span></a>, <a class="txt" href="#E15:128_5_2_9_3"><span class="lab"><font color="Green" title="E113">A140</font></span></a>, <a class="txt" href="#E14:128_5_2_9_3"><span class="lab"><font color="Green" title="E112">A139</font></span></a>, <a class="txt" href="#E12:128_5_2_9"><span class="lab"><font color="Green" title="E100">A127</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a>, <a class="txt" href="#E5:128_5_2_9_3"><span class="lab"><font color="Green" title="E105">A132</font></span></a>, <a class="txt" href="#E7:128_5_2_9_3"><span class="lab"><font color="Green" title="E107">A134</font></span></a>, <a class="txt" href="#E12:128_5_2_9_3"><span class="lab"><font color="Green" title="E111">A138</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">thus </span><font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<font color="Maroon" title="c10">H1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:128_5_2_9"><span class="lab"><font color="Green" title="E103">A130</font></span></a>, <a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E10:128_5_2_9"><span class="lab"><font color="Green" title="E99">A126</font></span></a>, <a class="txt" href="#E12:128_5_2_9"><span class="lab"><font color="Green" title="E100">A127</font></span></a>, <a class="txt" href="#E13:128_5_2_9"><span class="lab"><font color="Green" title="E101">A128</font></span></a>, <a class="txt" href="#E14:128_5_2_9"><span class="lab"><font color="Green" title="E102">A129</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c10">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_5_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</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="E19:128_5_2"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</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></div><span class="kw">end;</span></div>

</div>
