<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be    <a href="altcat_1.html#L1" title="ALTCAT_1:struct.1">AltGraph</a> ;<br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>thus </b><a NAME="E1:43"/>
( <font color="Maroon" title="c2">o</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a> implies for <font color="Olive" title="b1">o1</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">M</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font> st <br/>( <font color="Olive" title="b2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; ( for <font color="Olive" title="b3">M1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font>  st <font color="Olive" title="b3">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M1</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">o</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">o1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">M</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font><b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c4">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:43_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:43_1"/><i><font color="Green" title="E27">A3</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E3:43_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E6:43_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E5:43_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:43_1"/><b>then </b>
for <font color="Olive" title="b1">M1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font>  st <font color="Olive" title="b1">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:43_1"/>
 ex <font color="Olive" title="b1">M</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font> st <br/>( <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; ( for <font color="Olive" title="b2">M1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font>  st <font color="Olive" title="b2">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M1</font> ) )
 <b>by </b><i><a class="txt" href="altcat_3.html#E3:43_1"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:43"/><i><font color="Green" title="E25">A4</font></i>: 
for <font color="Olive" title="b1">o1</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">M</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font> st <br/>( <font color="Olive" title="b2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; ( for <font color="Olive" title="b3">M1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font>  st <font color="Olive" title="b3">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M1</font> ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">o1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">ALTCAT_3:def 9</a><br/>

<b>consider </b><font color="Maroon" title="c4">M</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font><b> such that </b><br/><a NAME="E4:43"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E5:43"/><i><font color="Green" title="E27">A6</font></i>: 
for <font color="Olive" title="b1">M1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font>  st <font color="Olive" title="b1">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M1</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:43"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green" title="E28">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">M1</font> = <font color="Maroon" title="c5">x</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font> ;<br/>
<a NAME="E3:43_2"/>
<font color="Maroon" title="c6">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E5:43"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="altcat_3.html#E1:43_2"><i><font color="Green" title="E29">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_2"/>
<font color="Maroon" title="c5">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="c4">M</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><b>end;</b></div>
<a NAME="E7:43"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_3"/>
<font color="Maroon" title="c5">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="c4">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:43_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:43"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:43"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E6:43"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:43"/>
 ex <font color="Olive" title="b1">M</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font> st <br/>( <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 ;<br/>


</div>
