<?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">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><br/>  for <font color="Olive" title="b3">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) holds <br/>( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b2">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</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="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b3">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b3">f</font>,<span class="p2">(<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></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">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">  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>  st <font color="Olive" title="b1">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="c2">a</font>,<font color="Maroon" title="c3">b</font>) holds <br/>( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</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="c3">b</font></span>)</span>,<font color="Olive" title="b1">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">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="c2">a</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c3">b</font>) implies ( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</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="c3">b</font></span>)</span>,<font color="Maroon" title="c4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">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="c2">a</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:73"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c3">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"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</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="c3">b</font></span>)</span>,<font color="Maroon" title="c4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">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="c2">a</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<a NAME="E2:73"/><span class="kw">then </span>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<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/>
<a NAME="E3:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c4">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="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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/>
<a NAME="E4:73"/>
 <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E37">A1</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="E5:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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/>
<a NAME="E6:73"/>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">id</a> <font color="Maroon" title="c2">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> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T27" onmouseover="rs('cat_1/T27')" onmouseout="rh()">CAT_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:73"/>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</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="c3">b</font></span>)</span>,<font color="Maroon" title="c4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">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="c2">a</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="catalg_1.html#T31" target="_self" onmouseover="rs('catalg_1/T31')" onmouseout="rh()">Th31</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>
