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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:75"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R2" target="_self">is_a_product_wrt</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E2:75"/><i><font color="Green">E42</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">f</font> is <a href="cat_1.html#V5">invertible</a> )
 ;<br/>

<a NAME="E3:75"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:75"/><b>then </b>
( <font color="Maroon">c</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:75"/><b>then </b>
<font color="Maroon">d</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="cat_3.html#K7" target="_self">*</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:75"/><b>then </b>
<font color="Maroon">d</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:75"/>
<font color="Maroon">d</font> <a href="cat_3.html#R2" target="_self">is_a_product_wrt</a> <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>,<font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


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