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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><br/>  for <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</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">g1</font> &amp; <font color="Olive" title="b3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">g2</font> holds <br/><span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b2">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f2</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font>  st <font color="Maroon" title="c3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Olive" title="b1">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/><span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Olive" title="b2">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c6">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">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="c3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g2</font> implies <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:89"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</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="c5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g2</font> or <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:89"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</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"> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">G1</font> =  <a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">G2</font> =  <a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">g2</font>);<br/>
<a NAME="E3:89"/>
<font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font>, <a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>), <a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>) <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T48" target="_self" onmouseover="rs('cat_8/T48')" onmouseout="rh()">Th48</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">H</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_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:89"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
( <font color="Maroon" title="c9">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font>) &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">g2</font>) )
 <span class="kw">and </span><br/><a NAME="E6:89"/>
 for <font color="Olive" title="b1">H1</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_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font>) &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">g2</font>) holds <br/><font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D17" target="_self" onmouseover="rs('cat_8/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g3</font>, <font color="Maroon" title="c11">f3</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_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:89"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
(  not <font color="Maroon" title="c10">g3</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  not <font color="Maroon" title="c11">f3</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c11">f3</font> )
 <span class="kw">and </span><br/><a NAME="E9:89"/>
 <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g3</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g3</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c11">f3</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E10:89"/>
 <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g3</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g3</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c11">f3</font></span>)</span>,<font color="Maroon" title="c10">g3</font>,<font color="Maroon" title="c11">f3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">f3</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c10">g3</font></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">and </span><br/><a NAME="E11:89"/>
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g3</font>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g3</font>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c11">f3</font>,<font color="Maroon" title="c10">g3</font>,<font color="Maroon" title="c11">f3</font>,<font color="Maroon" title="c11">f3</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c10">g3</font> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T18" target="_self" onmouseover="rs('cat_8/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E12:89"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<font color="Maroon" title="c11">f3</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">g3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="cat_7.html#T5" onmouseover="rs('cat_7/T5')" onmouseout="rh()">CAT_7:5</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A6</font></span>: <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c3">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="cat_8.html#D1" target="_self" onmouseover="rs('cat_8/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">g2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="cat_8.html#D1" target="_self" onmouseover="rs('cat_8/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<a NAME="E15:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>;<br/>
<span class="lab"><font color="Green" title="E55">A8</font></span>: <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c4">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="cat_8.html#D1" target="_self" onmouseover="rs('cat_8/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K1" title="CAT_8:func.1">COMPOSITION</a> (<font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">g2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c6">g2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="cat_8.html#D1" target="_self" onmouseover="rs('cat_8/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<a NAME="E18:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A9</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:89"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>;<br/>
<a NAME="E19:89"/>
<font color="Maroon" title="c9">H</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:89"/>
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:89"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E15:89"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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