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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="cat_6.html#V9" title="CAT_6:attr.9">composable</a>   <a href="cat_6.html#V10" title="CAT_6:attr.10">with_identities</a>   <a href="cat_6.html#L1" title="CAT_6:struct.1">CategoryStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> is  <a href="cat_8.html#V1" title="CAT_8:attr.1">terminal</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="cat_8.html#V1" title="CAT_8:attr.1">terminal</a>  holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> is  <a href="cat_8.html#V1" title="CAT_8:attr.1">terminal</a>  &amp; <font color="Maroon" title="c3">b</font> is  <a href="cat_8.html#V1" title="CAT_8:attr.1">terminal</a>  implies <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">a</font> is  <a href="cat_8.html#V1" title="CAT_8:attr.1">terminal</a> 
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c3">b</font> is  <a href="cat_8.html#V1" title="CAT_8:attr.1">terminal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">g</font> =  the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<a NAME="E3:23"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/>
<a NAME="E4:23"/>
 the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E2:23_1"/>
(  the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a>  the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font> &amp;  the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a>  the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="cat_8.html#T20" target="_self" onmouseover="rs('cat_8/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1"/>
 the    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="cat_7.html#D9" onmouseover="rs('cat_7/D9')" onmouseout="rh()">CAT_7:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:23"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D10" onmouseover="rs('cat_7/D10')" onmouseout="rh()">CAT_7:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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