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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM2" title="CAT_4:NM.2">Cocartesian_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> holds  <font color="Olive" title="b1">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b2">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Olive" title="b1">a</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> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> </span><br/>





<a NAME="E1:100"/><i><font color="Green" title="E48">A1</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</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#T66" target="_self" title="CAT_4:th.66">Th66</a></i>;<br/>
<b>hence </b><a NAME="E2:100"/><i><font color="Green" title="E49">A2</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</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#T70" target="_self" title="CAT_4:th.70">Th70</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D2" target="_self" title="CAT_4:def.2">CAT_4:def 2</a> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</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>  &amp;  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> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span> ) )</span><br/>

<a NAME="E3:100"/><i><font color="Green" title="E50">A3</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</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#T66" target="_self" title="CAT_4:th.66">Th66</a></i>;<br/>
<b>hence </b><a NAME="E4:100"/><i><font color="Green" title="E51">A4</font></i>: 
 <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="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</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#T70" target="_self" title="CAT_4:th.70">Th70</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">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span> )</span><br/>

<b>take </b><font color="Maroon" title="c4">f'</font> = <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>; <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="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font> st <br/>( <font color="Maroon" title="c4">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b1">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span> )</span><br/>

<b>take </b><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>; <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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span> )</span><br/>

<b>thus </b><font color="Maroon" title="c4">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
<b>by </b><i><a class="txt" href="#E2:100"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E3:100"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self" title="CAT_4:th.72">Th72</a></i>
<br/>.= 
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
<b>by </b><i><a class="txt" href="#E1:100"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a></i>
<br/>.= 
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
<b>by </b><i><a class="txt" href="#E1:100"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_4.html#T71" target="_self" title="CAT_4:th.71">Th71</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/>

<b>thus </b><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f'</font> = 
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
<b>by </b><i><a class="txt" href="#E1:100"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E4:100"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self" title="CAT_4:th.72">Th72</a></i>
<br/>.= 
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
<b>by </b><i><a class="txt" href="#E3:100"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a></i>
<br/>.= 
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
<b>by </b><i><a class="txt" href="#E3:100"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_4.html#T71" target="_self" title="CAT_4:th.71">Th71</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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