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

<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">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">F1</font>, <font color="Olive" title="b2">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b3">G1</font>, <font color="Olive" title="b4">G2</font>, <font color="Olive" title="b5">T</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Olive" title="b1">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b2">F2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">G2</font> &amp; <font color="Olive" title="b5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b4">G2</font> holds <br/><span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Olive" title="b5">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">T</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Maroon" title="c3">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b3">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> holds <br/><span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Olive" title="b3">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font>, <font color="Maroon" title="c7">T</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c2">D</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="c3">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">G2</font> &amp; <font color="Maroon" title="c7">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> implies <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c7">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( <font color="Maroon" title="c3">F1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">G2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> or <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c7">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:98"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c7">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</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="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c7">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><br/>

<a NAME="E3:98"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 <a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(#  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>, the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D34" onmouseover="rs('cat_6/D34')" onmouseout="rh()">CAT_6:def 34</a></span>;<br/>
<a NAME="E4:98"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 <a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(#  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>, the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D34" onmouseover="rs('cat_6/D34')" onmouseout="rh()">CAT_6:def 34</a></span>;<br/>
<a NAME="E5:98"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Olive" title="b1">a</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="98_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <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="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c8">a</font></span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c8">a</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</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="E2:98_1"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
<font color="Maroon" title="c9">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="ref" href="cat_6.html#T41" onmouseover="rs('cat_6/T41')" onmouseout="rh()">CAT_6:41</a></span>;<br/>
<a NAME="E3:98_1"/>
<font color="Maroon" title="c9">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T24" onmouseover="rs('cat_6/T24')" onmouseout="rh()">CAT_6:24</a>, <a class="ref" href="cat_6.html#T41" onmouseover="rs('cat_6/T41')" onmouseout="rh()">CAT_6:41</a></span>;<br/>
<a NAME="E4:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A7</font></span>: 
( <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">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="c5">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font></span>)</span> &amp; <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">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="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E55">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E57">A4</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="E6:98_1"/>
( <font color="Maroon" title="c5">G1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c6">G2</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="#E1:98"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="cat_6.html#T42" onmouseover="rs('cat_6/T42')" onmouseout="rh()">CAT_6:42</a></span>;<br/>
<a NAME="E7:98_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">G1</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a>  &amp; <font color="Maroon" title="c6">G2</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#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E8:98_1"/><span class="kw">then </span>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1"><span class="lab"><font color="Green" title="E60">A7</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>, <a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E59">A6</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="E9:98_1"/><span class="kw">then </span>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E54">A1</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="E10:98_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> &amp; <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T14" target="_self" onmouseover="rs('cat_8/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E11:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A8</font></span>: 
(  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c8">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T70" onmouseover="rs('cat_1/T70')" onmouseout="rh()">CAT_1:70</a></span>;<br/>
<a NAME="E12:98_1"/>
<font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T1" onmouseover="rs('cat_1/T1')" onmouseout="rh()">CAT_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:98_1"/>
<font color="Maroon" title="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c8">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_1"><span class="lab"><font color="Green" title="E61">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="E6:98"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  holds  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Olive" title="b1">a</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/>
<a NAME="E7:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" onmouseover="rs('nattra_1/D2')" onmouseout="rh()">NATTRA_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">T1</font> = <font color="Maroon" title="c7">T</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E57">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">t1</font> = <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c8">T1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<a NAME="E10:98"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
 ex <font color="Olive" title="b1">t</font> being    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c8">T1</font> &amp; (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b4">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b2">a</font></span>)</span> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="98_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:98_2"/>
 for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="98_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <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="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c10">a</font></span><br/>

<a NAME="E1:98_2_1"/>
<font color="Maroon" title="c10">a</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>
 
;<br/>
<a NAME="E2:98_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> = 
<font color="Maroon" title="c8">T1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_2_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c10">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_1.html#D12" onmouseover="rs('isocat_1/D12')" onmouseout="rh()">ISOCAT_1:def 12</a></span>
;<br/>
<a NAME="E4:98_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c10">a</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:98_2_1"/>
<font color="Maroon" title="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D5" onmouseover="rs('cat_1/D5')" onmouseout="rh()">CAT_1:def 5</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c9">t1</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="nattra_1.html#D3" onmouseover="rs('nattra_1/D3')" onmouseout="rh()">NATTRA_1:def 3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">t</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">t</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c8">T1</font> &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b3">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:98_2"/>
<font color="Maroon" title="c10">t</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c8">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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b3">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <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"> (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:98_2"/><span class="lab"><font color="Green" title="E62">A13</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">f</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span><br/>

<a NAME="E5:98_2"/>
<font color="Maroon" title="c11">a</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>
 
;<br/>
<a NAME="E6:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A14</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A15</font></span>: <font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font> = 
<font color="Maroon" title="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="nattra_1.html#D5" onmouseover="rs('nattra_1/D5')" onmouseout="rh()">NATTRA_1:def 5</a>, <a class="ref" href="nattra_1.html#D2" onmouseover="rs('nattra_1/D2')" onmouseout="rh()">NATTRA_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c8">T1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98_2"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_1.html#D12" onmouseover="rs('isocat_1/D12')" onmouseout="rh()">ISOCAT_1:def 12</a></span>
;<br/>
<a NAME="E8:98_2"/>
<font color="Maroon" title="c12">b</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>
 
;<br/>
<a NAME="E9:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A17</font></span>: <font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font> = 
<font color="Maroon" title="c9">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="nattra_1.html#D5" onmouseover="rs('nattra_1/D5')" onmouseout="rh()">NATTRA_1:def 5</a>, <a class="ref" href="nattra_1.html#D2" onmouseover="rs('nattra_1/D2')" onmouseout="rh()">NATTRA_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c8">T1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98_2"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_1.html#D12" onmouseover="rs('isocat_1/D12')" onmouseout="rh()">ISOCAT_1:def 12</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">g2</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c11">a</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</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">reconsider </span><font color="Maroon" title="c15">g1</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c12">b</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</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">reconsider </span><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c13">f</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</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="E14:98_2"/><span class="lab"><font color="Green" title="E67">A18</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="cat_1.html#D5" onmouseover="rs('cat_1/D5')" onmouseout="rh()">CAT_1:def 5</a></span>;<br/>
<a NAME="E15:98_2"/>
 <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c12">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="cat_1.html#T1" onmouseover="rs('cat_1/T1')" onmouseout="rh()">CAT_1:1</a></span>;<br/>
<a NAME="E16:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A19</font></span>: 
 <a href="cat_8.html#NK13" title="CAT_8:NK.13">KuratowskiPair</a> (<font color="Maroon" title="c15">g1</font>,<font color="Maroon" title="c16">g</font>) <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 <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<a NAME="E17:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A20</font></span>: 
<font color="Maroon" title="c15">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c16">g</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>;<br/>
<a NAME="E18:98_2"/>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="cat_1.html#T1" onmouseover="rs('cat_1/T1')" onmouseout="rh()">CAT_1:1</a></span>;<br/>
<a NAME="E19:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A21</font></span>: 
 <a href="cat_8.html#NK13" title="CAT_8:NK.13">KuratowskiPair</a> (<font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c14">g2</font>) <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 <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<a NAME="E20:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A22</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">g2</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>;<br/>
<a NAME="E21:98_2"/><span class="lab"><font color="Green" title="E72">A23</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Maroon" title="c15">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c15">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="98_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</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="c15">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> implies <font color="Maroon" title="c15">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_2_4"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
<font color="Maroon" title="c15">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c17">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</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="E3:98_2_4"/><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c12">b</font></span>)</span>,<font color="Maroon" title="c18">f</font></span><a href="cat_2.html#K8" title="CAT_2:func.8">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E1:98_2_4"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<a NAME="E4:98_2_4"/><span class="kw">then </span>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T15" onmouseover="rs('cat_1/T15')" onmouseout="rh()">CAT_1:15</a></span>;<br/>
<a NAME="E5:98_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T21" onmouseover="rs('cat_1/T21')" onmouseout="rh()">CAT_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:98_2_4"/>
<font color="Maroon" title="c15">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_2_4"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="ref" href="cat_6.html#T40" onmouseover="rs('cat_6/T40')" onmouseout="rh()">CAT_6:40</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="E22:98_2"/><span class="lab"><font color="Green" title="E73">A26</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">g2</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="98_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c1">C</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="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">g2</font> implies <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_2_5"/><span class="lab"><font color="Green" title="E74">A27</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">g2</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="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c17">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</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="E3:98_2_5"/><span class="lab"><font color="Green" title="E75">A28</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><font color="Maroon" title="c18">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="cat_2.html#K8" title="CAT_2:func.8">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E1:98_2_5"><span class="lab"><font color="Green" title="E74">A27</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<a NAME="E4:98_2_5"/><span class="kw">then </span>
 <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T15" onmouseover="rs('cat_1/T15')" onmouseout="rh()">CAT_1:15</a></span>;<br/>
<a NAME="E5:98_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T22" onmouseover="rs('cat_1/T22')" onmouseout="rh()">CAT_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:98_2_5"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_2_5"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="ref" href="cat_6.html#T40" onmouseover="rs('cat_6/T40')" onmouseout="rh()">CAT_6:40</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="E23:98_2"/><span class="lab"><font color="Green" title="E74">A29</font></span>: 
( <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">g2</font> &amp; <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g1</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">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font></span>)</span> &amp; <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</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="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">g2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98_2"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="txt" href="#E20:98_2"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E55">A2</font></span></a></span>;<br/>
<a NAME="E24:98_2"/><span class="lab"><font color="Green" title="E75">A30</font></span>: 
( <font color="Maroon" title="c15">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> &amp; <font color="Maroon" title="c16">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98_2"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E19:98_2"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E21:98_2"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E22:98_2"><span class="lab"><font color="Green" title="E73">A26</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<a NAME="E25:98_2"/><span class="lab"><font color="Green" title="E76">A31</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c12">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c12">b</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/>
<a NAME="E26:98_2"/><span class="lab"><font color="Green" title="E77">A32</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c12">b</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="cat_1.html#T82" onmouseover="rs('cat_1/T82')" onmouseout="rh()">CAT_1:82</a></span>;<br/>
<a NAME="E27:98_2"/><span class="lab"><font color="Green" title="E78">A33</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/>
<a NAME="E28:98_2"/><span class="lab"><font color="Green" title="E79">A34</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K8" title="CAT_1:func.8">.</a> <font color="Maroon" title="c12">b</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="cat_1.html#T82" onmouseover="rs('cat_1/T82')" onmouseout="rh()">CAT_1:82</a></span>;<br/>
<a NAME="E29:98_2"/><span class="lab"><font color="Green" title="E80">A35</font></span>: 
<font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98_2"><span class="lab"><font color="Green" title="E66">A17</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="E30:98_2"/><span class="lab"><font color="Green" title="E81">A36</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E54">A1</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="E31:98_2"/><span class="lab"><font color="Green" title="E82">A37</font></span>: 
<font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98_2"><span class="lab"><font color="Green" title="E64">A15</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="E32:98_2"/><span class="lab"><font color="Green" title="E83">A38</font></span>: 
<font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E54">A1</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="E33:98_2"/><span class="lab"><font color="Green" title="E84">A39</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span></span><a href="cat_2.html#K8" title="CAT_2:func.8">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:98_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="txt" href="#E30:98_2"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E23:98_2"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<a NAME="E34:98_2"/><span class="lab"><font color="Green" title="E85">A40</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="cat_2.html#K8" title="CAT_2:func.8">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:98_2"><span class="lab"><font color="Green" title="E82">A37</font></span></a>, <a class="txt" href="#E32:98_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E23:98_2"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Maroon" title="c13">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Maroon" title="c13">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98_2"><span class="lab"><font color="Green" title="E76">A31</font></span></a>, <a class="txt" href="#E26:98_2"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="ref" href="cat_1.html#D13" onmouseover="rs('cat_1/D13')" onmouseout="rh()">CAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="cat_3.html#D10" onmouseover="rs('cat_3/D10')" onmouseout="rh()">CAT_3:def 10</a></span>
<br/>.= 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c12">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:98_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>
<br/>.= 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:98_2"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E10:98_2"><span class="lab"><font color="Green" title="E66">A17</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">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g1</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">G1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:98_2"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="cat_6.html#D3" onmouseover="rs('cat_6/D3')" onmouseout="rh()">CAT_6:def 3</a></span>
<br/>.= 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <span class="p1">(<span class="default"><a href="cat_6.html#K18" title="CAT_6:func.18">alter</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c16">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">g2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:98_2"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="txt" href="#E23:98_2"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="ref" href="cat_6.html#D3" onmouseover="rs('cat_6/D3')" onmouseout="rh()">CAT_6:def 3</a></span>
<br/>.= 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:98_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E7:98_2"><span class="lab"><font color="Green" title="E64">A15</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">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:98_2"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="cat_3.html#D10" onmouseover="rs('cat_3/D10')" onmouseout="rh()">CAT_3:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:98_2"><span class="lab"><font color="Green" title="E78">A33</font></span></a>, <a class="txt" href="#E28:98_2"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="ref" href="cat_1.html#D13" onmouseover="rs('cat_1/D13')" onmouseout="rh()">CAT_1:def 13</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="E11:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A41</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="nattra_1.html#D7" onmouseover="rs('nattra_1/D7')" onmouseout="rh()">NATTRA_1:def 7</a>, <a class="ref" href="nattra_1.html#D2" onmouseover="rs('nattra_1/D2')" onmouseout="rh()">NATTRA_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">t</font> being    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font><span class="kw"> such that </span><br/><a NAME="E13:98"/><span class="lab"><font color="Green" title="E63">A42</font></span>: 
( <font color="Maroon" title="c10">t</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c8">T1</font> &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b3">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_3.html#K9" title="CAT_3:func.9">/.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K2" title="NATTRA_1:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98"><span class="lab"><font color="Green" title="E61">A11</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:98"/>
<span class="p1">(<span class="default"><a href="isocat_1.html#K7" title="ISOCAT_1:func.7">IdMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c7">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E62">A41</font></span></a>, <a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="ref" href="nattra_1.html#D8" onmouseover="rs('nattra_1/D8')" onmouseout="rh()">NATTRA_1:def 8</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>
