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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_6.html#V9" title="CAT_6:attr.9">composable</a>   <a href="cat_6.html#V10" title="CAT_6:attr.10">with_identities</a>   <a href="cat_6.html#L1" title="CAT_6:struct.1">CategoryStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> is  <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  implies <font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V8" title="CAT_6:attr.8">associative</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_1_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c1">C</font> is  <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V8" title="CAT_6:attr.8">associative</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:53_1_1_1"/>
( <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_1"/>
<font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V8" title="CAT_6:attr.8">associative</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:53_1_1_1_1"/>
<font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V8" title="CAT_6:attr.8">associative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V8" title="CAT_6:attr.8">associative</a> </span><br/><div class="add"><a NAME="E2:53_1_1_1_2"/>
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Olive" title="b2">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f3</font> &amp; <font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f3</font> &amp; <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">f3</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">f3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">f3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font>, <font color="Maroon" title="c4">f3</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">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="c2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">f3</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">f3</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font> implies <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_1_1_1_2_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
( <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">f3</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">f3</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</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="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">f11</font> = <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f22</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font>;<br/>
<span class="lab"><font color="Green" title="E34">A4</font></span>:  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span></span>)</span> = 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c4">f3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span></span>)</span> = 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_7.html#T4" target="_self" onmouseover="rs('cat_7/T4')" onmouseout="rh()">Th4</a></span>
;<br/>
<a NAME="E4:53_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span></span>)</span></span>)</span>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span></span>)</span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1_1_2_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E1:53_1_1_1_2"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="cat_7.html#T19" target="_self" onmouseover="rs('cat_7/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:53_1_1_1_2_1"/>
<font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:53_1_1_1_2"/>
<font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V8" title="CAT_6:attr.8">associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D10" onmouseover="rs('cat_6/D10')" onmouseout="rh()">CAT_6:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
