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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="index_1.html#NM9">Indexing</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="cat_5.html#V3">Categorial</a> <a href="cat_1.html#NM4">Category</a>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E23</font></i>: 
 <a href="index_1.html#K16" target="_self">rng</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>thus </b><a NAME="E2:48_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D9" target="_self">INDEX_1:def 9</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:48_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="index_1.html#K16" target="_self">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="index_1.html#D9" target="_self">Def9</a></i>;<br/><b>hence </b><a NAME="E2:48_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="index_1.html#E1:48_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="index_1.html#K16" target="_self">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="index_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E3:48_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <span class="p4">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> 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="index_1.html#E1:48_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cat_2.html#T14">CAT_2:14</a></i>;<br/>


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

<b>assume </b><a NAME="E2:48"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="index_1.html#M4" target="_self">TargetCat</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:48"/>
 <a href="index_1.html#K16" target="_self">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_5.html#K7">Image</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="index_1.html#K15" target="_self">-functor</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="index_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E5:48"/>
 <a href="index_1.html#K16" target="_self">rng</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
