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<b>let </b><font color="Maroon" title="c1">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="c1">C</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st  <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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Olive" title="b1">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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"> for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<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>  holds <br/><font color="Olive" title="b1">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span><br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<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>  implies <font color="Maroon" title="c4">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  )</span><br/>





<b>assume </b><a NAME="E1:10"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span><br/>

<a NAME="E2:10"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<a NAME="E3:10"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13" target="_self" title="OPPCAT_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:10"/>
<font color="Maroon" title="c4">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">a</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</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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