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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> (  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font>) &amp;  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">a</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> (  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">f</font>) &amp;  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">a</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>) &amp;  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>) )</span><br/>





<a NAME="E1:76"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></span>;<br/>
<a NAME="E2:76"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:76_1"/>
 <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="c3">f</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T2" title="CAT_1:th.2">CAT_1:2</a></span>;<br/><a NAME="E2:76_1"/><span class="kw">then </span>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<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>) <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> (<font color="Maroon" title="c2">a</font>,<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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T24" title="CAT_1:th.24">CAT_1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>)</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> (<font color="Maroon" title="c2">a</font>,<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>) &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>)</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> (<font color="Maroon" title="c2">a</font>,<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="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> (<font color="Maroon" title="c2">a</font>,<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>) 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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>)</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"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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> (<font color="Maroon" title="c2">a</font>,<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>) 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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E4:76_1"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon" title="c4">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> (<font color="Maroon" title="c2">a</font>,<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">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="E6:76_1"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/><a NAME="E7:76_1"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/><span class="kw">thus </span><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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<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="c5">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="ens_1.html#D23" target="_self" title="ENS_1:def.23">Def24</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E6:76_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E7:76_1"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:76_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="ens_1.html#D19" target="_self" title="ENS_1:def.19">Def20</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:76"/>
 <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:76_2"/>
 <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="c3">f</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T2" title="CAT_1:th.2">CAT_1:2</a></span>;<br/><a NAME="E2:76_2"/><span class="kw">then </span>
(  <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>,<font color="Maroon" title="c2">a</font>) <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>,<font color="Maroon" title="c2">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 class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T24" title="CAT_1:th.24">CAT_1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_2"/>
(  <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> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</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>,<font color="Maroon" title="c2">a</font>) &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)</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>,<font color="Maroon" title="c2">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>,<font color="Maroon" title="c2">a</font>) holds <br/><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>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</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"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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>,<font color="Maroon" title="c2">a</font>) implies <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>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E4:76_2"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c4">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>,<font color="Maroon" title="c2">a</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">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="E6:76_2"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/><span class="kw">thus </span><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>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E4:76_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="ens_1.html#D23" target="_self" title="ENS_1:def.23">Def24</a></span>
<br/>.= 
<font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:76_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="cat_1.html#T21" title="CAT_1:th.21">CAT_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="ens_1.html#D20" target="_self" title="ENS_1:def.20">Def21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:76"/>
 <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> (<font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
