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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Olive" title="b1">C</font>  st  <a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b3">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></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="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><br/><b>let </b><font color="Maroon" title="c2">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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font>  st  <a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b2">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</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#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> implies <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></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="c3">a</font>,<font color="Maroon" title="c4">b</font> )</span><br/>







<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E63">A1</font></i>: 
 <a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 ; <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="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></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="c3">a</font>,<font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E2:91"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c2">C</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">A</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E1:91"><i><font color="Green" title="E63">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">d</font> =  <a href="ens_1.html#K13" title="ENS_1:func.13">@</a> <font color="Maroon" title="c5">A</font>;<br/>
<span class="p1">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T41" title="CAT_2:th.41">CAT_2:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T21" title="OPPCAT_1:th.21">OPPCAT_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><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="c3">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K23" title="ENS_1:func.23">hom??</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><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="c3">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="#E1:91"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="ens_1.html#D28" target="_self" title="ENS_1:def.28">Def28</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" title="ENS_1:func.13">@</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:91"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="ens_1.html#E92"><i><font color="Green" title="E58">Lm10</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:91"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K26" title="ENS_1:func.26">hom??</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></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="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T103" title="CAT_1:th.103">CAT_1:103</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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