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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM1" title="CAT_4:NM.1">Cartesian_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>, <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Olive" title="b3">c</font></span>)</span></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>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Olive" title="b3">c</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><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></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>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</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><br/>







<a NAME="E1:53"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E2:53"/><i><font color="Green" title="E27">A2</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</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> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E3:53"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:53"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>
<a NAME="E4:53"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E5:53"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_4.html#E2:53"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>
<a NAME="E6:53"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E7:53"/><b>then </b>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</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> 
 
<b>by </b><i><a class="txt" href="cat_4.html#E5:53"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="cat_4.html#T25" target="_self" title="CAT_4:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E8:53"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></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> 
 <b>by </b><i><a class="txt" href="cat_4.html#E3:53"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="cat_4.html#T25" target="_self" title="CAT_4:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</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><br/>

<a NAME="E9:53"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green" title="E31">A6</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</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> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E11:53"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_4.html#E9:53"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>
<a NAME="E12:53"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E13:53"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_4.html#E10:53"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>
<a NAME="E14:53"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E15:53"/><b>then </b>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</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> 
 
<b>by </b><i><a class="txt" href="cat_4.html#E11:53"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="cat_4.html#T25" target="_self" title="CAT_4:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E16:53"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c4">c</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> 
 <b>by </b><i><a class="txt" href="cat_4.html#E13:53"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="cat_4.html#T25" target="_self" title="CAT_4:th.25">Th25</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>
