<?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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font> is  <a href="altcat_1.html#V3" title="ALTCAT_1:attr.3">associative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c5">l</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_1.html#D7" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> or  not <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font> or  not <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c7">g</font></span>)</span>,<font color="Maroon" title="c6">f</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font> or  not <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c7">g</font></span>)</span>,<font color="Maroon" title="c6">f</font> )</span><br/>

<a NAME="E2:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f9</font> = <font color="Maroon" title="c6">f</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:28"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c7">g</font></span>)</span>,<font color="Maroon" title="c6">f</font> )</span><br/>

<a NAME="E5:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">g9</font> = <font color="Maroon" title="c7">g</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:28"/>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K3" title="ALTCAT_2:func.3">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c7">g</font></span>)</span>,<font color="Maroon" title="c6">f</font></span><br/>

<a NAME="E8:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">h9</font> = <font color="Maroon" title="c8">h</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<a NAME="E10:28"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h9</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">g9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="altcat_2.html#T14" target="_self" title="ALTCAT_2:th.14">Th14</a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></span>;<br/>
<a NAME="E11:28"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g9</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">f9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="altcat_2.html#T14" target="_self" title="ALTCAT_2:th.14">Th14</a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> = 
<font color="Maroon" title="c11">h9</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">f9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="altcat_2.html#T14" target="_self" title="ALTCAT_2:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">h9</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">g9</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="cat_1.html#T54" title="CAT_1:th.54">CAT_1:54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_comps_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">h</font>,<font color="Maroon" title="c7">g</font></span>)</span>,<font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="altcat_2.html#T14" target="_self" title="ALTCAT_2:th.14">Th14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
