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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g'</font>, <font color="Olive" title="b4">f'</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">f'</font> holds <br/> <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Olive" title="b1">g</font>,<font color="Olive" title="b3">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">f'</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g'</font>, <font color="Maroon" title="c5">f'</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f'</font> implies  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span> )</span><br/>









<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E56">A1</font></i>: 
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span><br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green" title="E57">A2</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">g'</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">g</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c6">h</font> =  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">h2</font> =  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font>;<br/><b>set </b><font color="Maroon" title="c8">h1</font> =  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font>;<br/><a NAME="E1:79_1"/><i><font color="Green" title="E58">A3</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">g'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">g'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="ens_1.html#T51" target="_self" title="ENS_1:th.51">Th51</a></i>;<br/><a NAME="E2:79_1"/><i><font color="Green" title="E59">A4</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="ens_1.html#T51" target="_self" title="ENS_1:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E3:79_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="#E1:79_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><b>thus </b><a NAME="E4:79_1"/><i><font color="Green" title="E60">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E1:79_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="#E2:79_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span> implies <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><b>assume </b><a NAME="E5:79_1"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/><a NAME="E7:79_1"/><i><font color="Green" title="E62">A7</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">g'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E5:79_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="ens_1.html#T51" target="_self" title="ENS_1:th.51">Th51</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">l</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/><a NAME="E9:79_1"/><i><font color="Green" title="E63">A8</font></i>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">f'</font>
 <b>by </b><i><a class="txt" href="#E5:79_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><a NAME="E10:79_1"/><i><font color="Green" title="E64">A9</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E5:79_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><a NAME="E11:79_1"/><b>then </b><i><font color="Green" title="E65">A10</font></i>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><a NAME="E12:79_1"/><i><font color="Green" title="E66">A11</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">k</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f'</font> )
 <b>by </b><i><a class="txt" href="#E9:79_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><a NAME="E13:79_1"/><b>then </b><i><font color="Green" title="E67">A12</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f'</font> )
 <b>by </b><i><a class="txt" href="#E10:79_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="#E5:79_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="ens_1.html#D24" target="_self" title="ENS_1:def.24">Def24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="#E9:79_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="#E10:79_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E9:79_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="#E11:79_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="#E9:79_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="#E10:79_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E10:79_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="#E12:79_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font>
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E13:79_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font>
<b>by </b><i><a class="txt" href="#E5:79_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="ens_1.html#D24" target="_self" title="ENS_1:def.24">Def24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font>
<b>by </b><i><a class="txt" href="#E7:79_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="ens_1.html#D24" target="_self" title="ENS_1:def.24">Def24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="txt" href="#E4:79_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="#E5:79_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
 <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g'</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c4">g'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">f'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
