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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>  st <font color="Olive" title="b1">a</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a> &amp; <font color="Olive" title="b2">b</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</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"> ( <font color="Maroon" title="c2">a</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a> &amp; <font color="Maroon" title="c3">b</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a> implies <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c2">a</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c3">b</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">terminal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> </span><br/>

<b>thus </b><a NAME="E3:76"/><i><font color="Green" title="E41">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">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> 
 <b>by </b><i><a class="txt" href="cat_1.html#E2:76"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="cat_1.html#D15" target="_self" title="CAT_1:def.15">Def15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" target="_self" title="CAT_1:def.17">CAT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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 <font color="Olive" title="b1">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">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>;<br/>
<b>consider </b><font color="Maroon" title="c5">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</font>
; <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> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:76_1"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<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_1.html#E1:76"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="cat_1.html#D15" target="_self" title="CAT_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Olive" title="b1">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>take </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font>; <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="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>thus </b><a NAME="E2:76_1"/>
( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:76"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:76"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="cat_1.html#T87" target="_self" title="CAT_1:th.87">Th87</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>
<b>hence </b><a NAME="E7:76"/>
<font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E3:76"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</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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