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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="cat_5.html#V2" target="_self">categorial</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_1.html#NM4">Category</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E25</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green">E27</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> iff <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E5:39"/>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>8</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self">Th12</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E5:39_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E6:39_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_5.html#E2:39"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_5.html#E5:39_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E7:39_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_5.html#E4:39"><i><font color="Green">E28</font></i></a>, <a class="txt" href="cat_5.html#E5:39_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E8:39_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>8</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E6:39_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E7:39_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_5.html#E4:39"><i><font color="Green">E28</font></i></a>, <a class="txt" href="cat_5.html#E6:39_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:39"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_5.html#E3:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_5.html#E6:39_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:39_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_5.html#E3:39"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cat_5.html#T14" target="_self">Th14</a></i>;<br/>


</div>
