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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM1" title="CAT_4:NM.1">Cartesian_category</a>;<br/>

<a NAME="E1:31"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>
 
<b>by </b><i><a class="ref" href="cat_4.html#T13" target="_self" title="CAT_4:th.13">Th13</a></i>;<br/>
<a NAME="E2:31"/>
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>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<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>;<br/>



<b>take </b>
<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font>
;<br/>

<b>take </b>
 <a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
;<br/>

<b>take </b>
 <a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
;<br/>

<b>thus </b><a NAME="E1:31_1"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a>  <a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="cat_4.html#T19" target="_self" title="CAT_4:th.19">Th19</a>, <a class="ref" href="cat_4.html#T20" target="_self" title="CAT_4:th.20">Th20</a>, <a class="ref" href="cat_4.html#T22" target="_self" title="CAT_4:th.22">Th22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
<font color="Maroon" title="c1">C</font> is <a href="cat_4.html#V1" title="CAT_4:attr.1">with_finite_product</a>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="cat_4.html#T1" target="_self" title="CAT_4:th.1">Th1</a></i>;<br/>


</div>
