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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">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="xboole_0.html#K1">{}</a> ,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>





<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon">a</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">F</font> implies <font color="Maroon">a</font> is <a href="cat_1.html#V6">terminal</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">F</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D15">CAT_1:def 15</a><br/>

<b>consider </b><font color="Maroon">F'</font> being    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">b</font>, <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E3:65_1"/>
(  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:65_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E6:65_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> holds <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>thus </b><a NAME="E7:65_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">h</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font> <b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font>;<br/>

<a NAME="E9:65_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:65_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:65"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<b>thus </b><a NAME="E3:65"/>
<font color="Maroon">F</font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">a</font>, <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D15" target="_self">CAT_3:def 15</a><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>let </b><font color="Maroon">F'</font> be    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">b</font>, <a href="xboole_0.html#K1">{}</a> ;<br/>



<b>assume </b><a NAME="E4:65"/>
 <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font>
 ;<br/>

<b>consider </b><font color="Maroon">h</font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font><b> such that </b><br/><a NAME="E6:65"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">g</font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font> holds  <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i><a class="ref" href="cat_3.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<a NAME="E7:65"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_3.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/>
<b>hence </b><a NAME="E8:65"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E9:65"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E10:65"/><b>then </b>
<font color="Maroon">k</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:65"/>
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>


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