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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> 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="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> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  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="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="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="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="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="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">a</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">thus </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="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<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 class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</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_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c3">a</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T21" title="OPPCAT_1:th.21">OPPCAT_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</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_1.html#K1" title="FUNCT_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="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <span class="p2">(<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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="ens_1.html#D28" target="_self" title="ENS_1:def.28">Def28</a></span>
<br/>.= 
 <a href="ens_1.html#K20" title="ENS_1:func.20">hom?-</a> <font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T57" target="_self" title="ENS_1:th.57">Th57</a></span>
<br/>.= 
 <a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="ens_1.html#D26" target="_self" title="ENS_1:def.26">Def26</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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