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

<b>consider </b><font color="Maroon">G0</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G0</font>,<font color="Maroon">G0</font>;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Olive">G</font>,<font color="Olive">H</font></span>)</span> where <font color="Olive">G</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Olive">H</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> : <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> </span>}</span> ;<br/>
<a NAME="E2:60_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">G0</font> is    <a href="ringcat1.html#M5" target="_self">Element</a> of  <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G0</font>,<font color="Maroon">G0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:60_1_1"/>
 <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G0</font>,<font color="Maroon">G0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Olive">G</font>,<font color="Olive">H</font></span>)</span> where <font color="Olive">G</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Olive">H</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> : <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">T</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Olive">G</font>,<font color="Olive">H</font></span>)</span> where <font color="Olive">G</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Olive">H</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> : <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:60_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff  ex <font color="Olive">G</font>, <font color="Olive">H</font> being   <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> &amp; <font color="Olive">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">G</font>,<font color="Olive">H</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:60_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> implies  ex <font color="Olive">G</font>, <font color="Olive">H</font> being   <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> &amp; <font color="Maroon">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">G</font>,<font color="Olive">H</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:60_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Olive">G</font>,<font color="Olive">H</font></span>)</span> where <font color="Olive">G</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Olive">H</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> : <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">H</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:60_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G</font>,<font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E7:60_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="ringcat1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E8:60_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> is    <a href="ringcat1.html#M5" target="_self">Element</a> of  <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G</font>,<font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E9"><i><font color="Green">Lemma39</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E9:60_1_1_1_1"/>
( <font color="Maroon">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>


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

<b>thus </b><a NAME="E2:60_1_1_1"/>
(  ex <font color="Olive">G</font>, <font color="Olive">H</font> being   <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> &amp; <font color="Maroon">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">G</font>,<font color="Olive">H</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">G</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">H</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:60_1_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E2:60_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 ;<br/>

<b>set </b><font color="Maroon">M</font> =  <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>
<a NAME="E3:60_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G</font>,<font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E9"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="ringcat1.html#D6" target="_self">Def6</a>, </i>;<br/>
<a NAME="E4:60_1_1_1_2"/>
 <a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" target="_self">Morphs</a> <font color="Olive">G</font>,<font color="Olive">H</font></span>)</span> where <font color="Olive">G</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Olive">H</font> is    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> : <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ringcat1.html#E9"><i><font color="Green">Lemma39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:60_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:60_1_1_2"/><b>then </b>
 ex <font color="Olive">G</font>, <font color="Olive">H</font> being   <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> &amp; <font color="Maroon">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">G</font>,<font color="Olive">H</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:60_1_1_2"/>
<font color="Maroon">x</font> is  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T'</font> = <font color="Maroon">T</font> as   <a href="ringcat1.html#NM5">RingMorphism_DOMAIN</a> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">T'</font>
;<br/>

<b>thus </b><a NAME="E8:60_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T'</font> iff  ex <font color="Olive">G</font>, <font color="Olive">H</font> being   <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">H</font> &amp; <font color="Olive">x</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">G</font>,<font color="Olive">H</font> ) )
 <b>by </b><i/>;<br/>


</div>
