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<b>let </b><font color="Maroon" title="c1">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font>;<br/>





<b>set </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span>;<br/>
<a NAME="E1:94"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E2:94"/>
<font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="cat_3.html#T69" target="_self" title="CAT_3:th.69">Th69</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">CAT_3:def 18</a><br/>

<b>let </b><font color="Maroon" title="c5">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font>;<br/><b>let </b><font color="Maroon" title="c6">F'</font> be    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c5">b</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>



<b>assume </b><a NAME="E3:94"/><i><font color="Green" title="E55">A1</font></i>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c6">F'</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">y</font>;<br/>

<a NAME="E4:94"/><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:94_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></i>;<br/><b>thus </b> <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c7">h</font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">y</font>
<b>by </b><i><a class="txt" href="cat_3.html#E3:94"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="cat_3.html#E4:94"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:94"/>
<font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E7:94"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>
 ;<br/>

<a NAME="E8:94"/><b>then </b><i><font color="Green" title="E57">A3</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<b>thus </b><a NAME="E9:94"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Maroon" title="c8">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:94_2"/><i><font color="Green" title="E58">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Maroon" title="c8">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c8">k</font> = 
<font color="Maroon" title="c8">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E8:94"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c8">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="cat_3.html#E1:94_2"><i><font color="Green" title="E58">A4</font></i></a></i>
;<br/>


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

<b>assume </b><a NAME="E10:94"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E11:94"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>hence </b><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">x</font> = 
<font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="cat_3.html#E10:94"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c8">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E8:94"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c8">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">y</font> <a href="cat_3.html#K2" title="CAT_3:func.2">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E11:94"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></i>
;<br/><br/>


</div>
