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

<span class="kw">let </span><font color="Maroon" title="c1">C</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">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">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">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> holds <br/>(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b2">g</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="Olive" title="b1">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b2">g</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="Olive" title="b1">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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">  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="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/>(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b1">g</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="c2">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">g</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="c2">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> implies (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:14"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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"> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span> )</span><br/>

<a NAME="E2:14"/><span class="lab"><font color="Green" title="E14">A2</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/>
<span class="kw">consider </span><font color="Maroon" title="c4">d1</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><span class="kw"> such that </span><br/><a NAME="E4:14"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d1</font> &amp; <font color="Maroon" title="c3">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">d1</font> &amp; <font color="Maroon" title="c4">d1</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#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">d11</font> = <font color="Maroon" title="c4">d1</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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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="E6:14"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c4">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <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="#E4:14"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <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#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<a NAME="E7:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c5">d11</font> <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="c5">d11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">d2</font> =  <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="c2">f</font></span>)</span> 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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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="E9:14"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c4">d1</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E10:14"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 for <font color="Olive" title="b1">f1</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">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">d2</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">f1</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="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">d2</font> implies <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">f11</font> = <font color="Maroon" title="c7">f1</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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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">assume </span><a NAME="E2:14_1"/>
<font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">d2</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">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font></span><br/>

<a NAME="E3:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><font color="Maroon" title="c8">f11</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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="E4:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A8</font></span>: 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">f11</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> <span class="p2">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">d2</font> = 
<font color="Maroon" title="c8">f11</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> <span class="p2">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="cat_6.html#T40" onmouseover="rs('cat_6/T40')" onmouseout="rh()">CAT_6:40</a></span>
<br/>.= 
<font color="Maroon" title="c7">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="cat_1.html#T22" onmouseover="rs('cat_1/T22')" onmouseout="rh()">CAT_1:22</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:14"/>
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">d11</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="#E1:14"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E15">A3</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="E12:14"/><span class="kw">then </span>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c5">d11</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> <span class="p2">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E16">A4</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="E13:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c6">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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/>
<span class="lab"><font color="Green" title="E20">A10</font></span>: <font color="Maroon" title="c4">d1</font> = 
<font color="Maroon" title="c4">d1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">d2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A6</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/>.= 
<font color="Maroon" title="c6">d2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E15:14"/>
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">c1</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><span class="kw"> such that </span><br/><a NAME="E17:14"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
(  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c7">c1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c7">c1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">c11</font> = <font color="Maroon" title="c7">c1</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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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="c9">c2</font> =  <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="c2">f</font></span>)</span> 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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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="E20:14"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c7">c1</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E21:14"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
 for <font color="Olive" title="b1">f1</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">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">c2</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f1</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="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">c2</font> implies <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">f11</font> = <font color="Maroon" title="c10">f1</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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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">assume </span><a NAME="E2:14_3"/>
<font color="Maroon" title="c10">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">c2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font></span><br/>

<a NAME="E3:14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><font color="Maroon" title="c11">f11</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></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="#E2:14"><span class="lab"><font color="Green" title="E14">A2</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="E4:14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A15</font></span>: 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c11">f11</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> <span class="p2">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c10">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">c2</font> = 
<font color="Maroon" title="c11">f11</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> <span class="p2">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_3"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="cat_6.html#T40" onmouseover="rs('cat_6/T40')" onmouseout="rh()">CAT_6:40</a></span>
<br/>.= 
<font color="Maroon" title="c10">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_3"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="cat_1.html#T22" onmouseover="rs('cat_1/T22')" onmouseout="rh()">CAT_1:22</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:14"/>
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><font color="Maroon" title="c8">c11</font>,<font color="Maroon" title="c2">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="#E1:14"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E21">A11</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="E23:14"/><span class="kw">then </span>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">c11</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> <span class="p2">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<a NAME="E24:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<span class="p1"><a href="cat_2.html#K8" title="CAT_2:func.8">[</a><span class="default"><font color="Maroon" title="c8">c11</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></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="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A17</font></span>: <font color="Maroon" title="c7">c1</font> = 
<font color="Maroon" title="c7">c1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c9">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E21:14"><span class="lab"><font color="Green" title="E23">A13</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/>.= 
<font color="Maroon" title="c9">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E26:14"/>
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c3">g</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="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E25:14"><span class="lab"><font color="Green" title="E25">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
