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

<span class="kw">set </span><font color="Maroon" title="c2">C</font> =  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">d</font> =  <a href="coh_sp.html#K25" title="COH_SP:func.25">fDom</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">c</font> =  <a href="coh_sp.html#K26" title="COH_SP:func.26">fCod</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">p</font> =  <a href="coh_sp.html#K27" title="COH_SP:func.27">fComp</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">thus </span><a NAME="E1:81_1_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V2" title="CAT_1:attr.2">Category-like</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"> (  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V3" title="CAT_1:attr.3">transitive</a>  &amp;  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">associative</a>  &amp;  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">reflexive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">ff</font>, <font color="Maroon" title="c8">gg</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</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"> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</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="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> or  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</font> ) &amp; (  not  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</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="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c7">ff</font>, <font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c8">gg</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E2:81_1_1_1"/>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K25" title="COH_SP:func.25">fDom</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c10">g</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K26" title="COH_SP:func.26">fCod</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c9">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a>, <a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:81_1_1_1"/>
( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</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="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> or  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</font> ) &amp; (  not  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</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="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</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">thus </span><a NAME="E2:81_1_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
 <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V3" title="CAT_1:attr.3">transitive</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"> (  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">associative</a>  &amp;  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">reflexive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">ff</font>, <font color="Maroon" title="c8">gg</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D7" onmouseover="rs('cat_1/D7')" onmouseout="rh()">CAT_1:def 7</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"> (  not  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</font> or (  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c7">ff</font> &amp;  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <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="c8">gg</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:81_1_1_2"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</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="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c7">ff</font> &amp;  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <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="c8">gg</font> )</span><br/>

<a NAME="E2:81_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</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_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/>
<a NAME="E3:81_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A4</font></span>: 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c7">ff</font>, <font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c8">gg</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E5:81_1_1_2"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="coh_sp.html#K25" title="COH_SP:func.25">fDom</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c10">g</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K26" title="COH_SP:func.26">fCod</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c9">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a>, <a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>;<br/>
<a NAME="E6:81_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<span class="p1">(<span class="default"><a href="coh_sp.html#K27" title="COH_SP:func.27">fComp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E7:81_1_1_2"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
( <span class="p1">(<span class="default"><a href="coh_sp.html#K25" title="COH_SP:func.25">fDom</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c9">f</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K26" title="COH_SP:func.26">fCod</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c10">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a>, <a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>;<br/>
<a NAME="E8:81_1_1_2"/>
(  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c9">f</font> &amp;  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c10">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_2"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E5:81_1_1_2"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" onmouseover="rs('coh_sp/T44')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:81_1_1_2"/>
(  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c7">ff</font> &amp;  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <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="c8">gg</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1_2"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E7:81_1_1_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a>, <a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a>, <a class="txt" href="#E3:81_1_1_2"><span class="lab"><font color="Green" title="E48">A4</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">thus </span><a NAME="E3:81_1_1"/>
 <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">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">  <a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">reflexive</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="81_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">ff</font>, <font color="Maroon" title="c8">gg</font>, <font color="Maroon" title="c9">hh</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D8" onmouseover="rs('cat_1/D8')" onmouseout="rh()">CAT_1:def 8</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"> (  not  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c9">hh</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="c8">gg</font> or  not  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</font> or <font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c8">gg</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:81_1_1_3"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c9">hh</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="c8">gg</font>
 <span class="kw">and </span><br/><a NAME="E2:81_1_1_3"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</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="c7">ff</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="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c8">gg</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c7">ff</font>, <font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c8">gg</font>, <font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c9">hh</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:81_1_1_3"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
(  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K25" title="COH_SP:func.25">fDom</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">h</font> &amp;  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K26" title="COH_SP:func.26">fCod</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a>, <a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>;<br/>
<a NAME="E5:81_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A11</font></span>: 
 <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" onmouseover="rs('coh_sp/T44')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E6:81_1_1_3"/><span class="lab"><font color="Green" title="E51">A12</font></span>: 
(  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K25" title="COH_SP:func.25">fDom</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> &amp;  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K26" title="COH_SP:func.26">fCod</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a>, <a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>;<br/>
<a NAME="E7:81_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A13</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c12">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E4:81_1_1_3"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" onmouseover="rs('coh_sp/T44')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E8:81_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">hh</font>,<font color="Maroon" title="c8">gg</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_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/>
<a NAME="E9:81_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c8">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">hh</font>,<font color="Maroon" title="c8">gg</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>;<br/>
<a NAME="E10:81_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</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_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/>
<a NAME="E11:81_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>;<br/>
<a NAME="E12:81_1_1_3"/>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c8">gg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c8">gg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a></span>;<br/>
<a NAME="E13:81_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c8">gg</font></span>)</span>,<font color="Maroon" title="c7">ff</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_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a></span>;<br/>
<a NAME="E14:81_1_1_3"/>
 <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> <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="c8">gg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a></span>;<br/>
<a NAME="E15:81_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">hh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span></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_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">gg</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font></span>)</span> = 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">hh</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">gg</font>,<font color="Maroon" title="c7">ff</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81_1_1_3"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="coh_sp.html#K27" title="COH_SP:func.27">fComp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c10">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E6:81_1_1_3"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>
<br/>.= 
<font color="Maroon" title="c12">h</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c10">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81_1_1_3"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E4:81_1_1_3"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="txt" href="#E6:81_1_1_3"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="coh_sp.html#T45" target="_self" onmouseover="rs('coh_sp/T45')" onmouseout="rh()">Th44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="coh_sp.html#K27" title="COH_SP:func.27">fComp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c11">g</font></span>)</span>,<font color="Maroon" title="c10">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E6:81_1_1_3"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="txt" href="#E5:81_1_1_3"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>
<br/>.= 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">hh</font>,<font color="Maroon" title="c8">gg</font>)</span>)</span>,<font color="Maroon" title="c7">ff</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_3"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E4:81_1_1_3"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">hh</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c8">gg</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c7">ff</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:81_1_1_3"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="txt" href="#E13:81_1_1_3"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>
;<br/> <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">let </span><font color="Maroon" title="c7">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D9" onmouseover="rs('cat_1/D9')" onmouseout="rh()">CAT_1:def 9</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">  not  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">aa</font> = <font color="Maroon" title="c7">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K16" title="COH_SP:func.16">TOL</a> <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ii</font> =  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c8">aa</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E47">A17</font></span>:  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c9">ii</font> = 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c8">aa</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>
<br/>.= 
<font color="Maroon" title="c8">aa</font>

;<br/>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c9">ii</font> = 
 <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c8">aa</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a></span>
<br/>.= 
<font color="Maroon" title="c8">aa</font>

;<br/>
<a NAME="E8:81_1_1"/><span class="kw">then </span>
 <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c8">aa</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="c7">a</font>,<font color="Maroon" title="c7">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1"><span class="lab"><font color="Green" title="E47">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:81_1_1"/>
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</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> 
 ; <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>
