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<b>let </b><font color="Maroon">h1</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span>, <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span>, <font color="Maroon">h2</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span>, <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds  <font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">g</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <font color="Olive">f</font>,<font color="Olive">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E2:82_1_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds  <font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">g</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <font color="Olive">f</font>,<font color="Olive">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:82_1_2_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>thus </b><font color="Maroon">h1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="ens_1.html#T19" target="_self">Th19</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_1_2"/>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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