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

<b>consider </b><font color="Maroon" title="c2">G0</font> being    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c2">G0</font>;<br/>
<b>set </b><font color="Maroon" title="c4">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font></span>)</span> where <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> is    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> </span>}</span> ;<br/>
<a NAME="E2:61_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
 <a href="ringcat1.html#K5" title="RINGCAT1:func.5">ID</a> <font color="Maroon" title="c2">G0</font> is    <a href="ringcat1.html#M5" title="RINGCAT1:mode.5">Element</a> of  <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c2">G0</font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#D15" target="_self" title="RINGCAT1:def.15">Def15</a></i>;<br/>
<a NAME="E3:61_1_1"/>
 <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c2">G0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font></span>)</span> where <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> is    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">T</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font></span>)</span> where <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> is    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="ringcat1.html#E2:61_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:61_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font> iff  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> ) )</span><br/>

<b>thus </b><a NAME="E1:61_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> ) implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_1_1_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>and </b><br/><a NAME="E4:61_1_1_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font></span>)</span> where <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> is    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">G</font>, <font color="Maroon" title="c9">H</font> being    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:61_1_1_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font>
 <b>and </b><br/><a NAME="E7:61_1_1_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E4:61_1_1_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<a NAME="E8:61_1_1_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M5" title="RINGCAT1:mode.5">Element</a> of  <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:61_1_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="ringcat1.html#E6:61_1_1_1_1"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c8">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G</font>,<font color="Olive" title="b1">H</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c9">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font> )</span><br/>

<b>thus </b><a NAME="E9:61_1_1_1_1"/>
( <font color="Maroon" title="c8">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E7:61_1_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="ringcat1.html#E8:61_1_1_1_1"><i><font color="Green" title="E43">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:61_1_1_1"/>
(  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> ) implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c7">G</font>, <font color="Maroon" title="c8">H</font> being    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><a NAME="E2:61_1_1_1_2"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c8">H</font>
 <b>and </b><br/><a NAME="E3:61_1_1_1_2"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font></span><br/>

<b>set </b><font color="Maroon" title="c9">M</font> =  <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">H</font>;<br/>
<a NAME="E4:61_1_1_1_2"/><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">H</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E2:61_1_1_1_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="ringcat1.html#E3:61_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="ringcat1.html#D15" target="_self" title="RINGCAT1:def.15">Def15</a></i>;<br/>
<a NAME="E5:61_1_1_1_2"/>
 <a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ringcat1.html#K8" title="RINGCAT1:func.8">Morphs</a> <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font></span>)</span> where <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> is    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ringcat1.html#E2:61_1_1_1_2"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:61_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E4:61_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font> implies <font color="Maroon" title="c6">x</font> is  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> )</span><br/><b>assume </b><a NAME="E1:61_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> is  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a></span><br/><a NAME="E2:61_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c6">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E5:61_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_2"/>
<font color="Maroon" title="c6">x</font> is  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">T'</font> = <font color="Maroon" title="c5">T</font> as   <a href="ringcat1.html#NM5" title="RINGCAT1:NM.5">RingMorphism_DOMAIN</a> <b>by </b><i><a class="ref" href="ringcat1.html#D13" target="_self" title="RINGCAT1:def.13">Def13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">T'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T'</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) )</span><br/>

<b>thus </b><a NAME="E8:61_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T'</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> ) )
 <b>by </b><i><a class="txt" href="ringcat1.html#E5:61_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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