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<b>let </b><font color="Maroon" title="c1">A</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">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>  st  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b4">m2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b3">m1</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Olive" title="b1">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="Olive" title="b3">m1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b4">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b4">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Olive" title="b2">g</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="Olive" title="b4">m2</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b4">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b3">m1</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>  st  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">m2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">m1</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Maroon" title="c2">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="Olive" title="b1">m1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Maroon" title="c3">g</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="Olive" title="b2">m2</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">m1</font></span><br/><b>let </b><font color="Maroon" title="c4">m1</font>, <font color="Maroon" title="c5">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Maroon" title="c2">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="c4">m1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Maroon" title="c3">g</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="c5">m2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span></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="c5">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">m1</font> )</span><br/>





<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E2">A1</font></i>: 
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Maroon" title="c2">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="c4">m1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<font color="Maroon" title="c3">g</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="c5">m2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span></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="c5">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">m1</font></span><br/>

<a NAME="E2:2"/><i><font color="Green" title="E3">A2</font></i>: 
 <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14" title="ENS_1:def.14">ENS_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">m1'</font> = <font color="Maroon" title="c4">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">m2'</font> = <font color="Maroon" title="c5">m2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E2:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<a NAME="E5:2"/><i><font color="Green" title="E4">A3</font></i>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/>
<i><font color="Green" title="E5">A4</font></i>:  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c6">m1'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
;<br/>
<a NAME="E7:2"/><i><font color="Green" title="E6">A5</font></i>: 
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E7">A6</font></i>:  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m1'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font> = 
<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">m2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E3:2_2"/>
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font> )
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:2_2"><i><font color="Green" title="E7">A6</font></i></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>
<a NAME="E8:2"/><i><font color="Green" title="E7">A7</font></i>: 
( <font color="Maroon" title="c7">m2'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c6">m1'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<a NAME="E1:2_3"/><i><font color="Green" title="E8">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m2'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m2'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">m2'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></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="c5">m2</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></i>;<br/>
<a NAME="E2:2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m1'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c6">m1'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">m1'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></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="c4">m1</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:2_3"/>
( <font color="Maroon" title="c7">m2'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c6">m1'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="yoneda_1.html#E1:2_3"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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>
<font color="Maroon" title="c5">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">m1</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">m2'</font>,<font color="Maroon" title="c6">m1'</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E5:2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c7">m2'</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c6">m1'</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E6:2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="ens_1.html#D12" title="ENS_1:def.12">ENS_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E6:2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="yoneda_1.html#E7:2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="yoneda_1.html#E8:2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="ens_1.html#D7" title="ENS_1:def.7">ENS_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E10:2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">m2</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span></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="c5">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">m1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
