<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="cat_4.html#NM2">Cocartesian_category</a>;<br/>

<a NAME="E1:88"/><i><font color="Green">E34</font></i>: 
 <a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">C</font> is <a href="cat_1.html#V7">initial</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:88"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ex <font color="Olive">i1</font>, <font color="Olive">i2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> st <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">i2</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">i2</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="cat_3.html#R4">is_a_coproduct_wrt</a> <font color="Olive">i1</font>,<font color="Olive">i2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>



<b>take </b>
<font color="Maroon">a</font> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">b</font>
;<br/>

<b>take </b>
 <a href="cat_4.html#K21" target="_self">in1</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
;<br/>

<b>take </b>
 <a href="cat_4.html#K22" target="_self">in2</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E1:88_1"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K21" target="_self">in1</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K22" target="_self">in2</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_4.html#K21" target="_self">in1</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_4.html#K22" target="_self">in2</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">b</font> <a href="cat_3.html#R4">is_a_coproduct_wrt</a>  <a href="cat_4.html#K21" target="_self">in1</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>, <a href="cat_4.html#K22" target="_self">in2</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:88"/>
<font color="Maroon">C</font> <a href="cat_4.html#NR3" target="_self">has_finite_coproduct</a> 
 <b>by </b><i>, </i>;<br/>


</div>
