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

<b>consider </b><font color="Maroon" title="c2">G0</font>, <font color="Maroon" title="c3">H0</font> being  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c4">M</font> =  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c3">H0</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">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="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : verum </span>}</span> ;<br/>
<a NAME="E2:83_1_1"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="grcat_1.html#K14" title="GRCAT_1:func.14">ZERO</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c3">H0</font> is    <a href="grcat_1.html#M7" title="GRCAT_1:mode.7">Element</a> of  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c3">H0</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D27" target="_self" title="GRCAT_1:def.27">Def27</a></i>;<br/>
<a NAME="E3:83_1_1"/>
 <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c2">G0</font>,<font color="Maroon" title="c3">H0</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="grcat_1.html#K18" title="GRCAT_1:func.18">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="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">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="grcat_1.html#K18" title="GRCAT_1:func.18">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="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : verum </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="grcat_1.html#E2:83_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:83_1_1"/><i><font color="Green" title="E47">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="c6">T</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1: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="83_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:83_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Maroon" title="c7">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_1"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_1"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c8">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="grcat_1.html#K18" title="GRCAT_1:func.18">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="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : verum </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="c9">G</font>, <font color="Maroon" title="c10">H</font> being  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:83_1_1_1_1"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E4:83_1_1_1_1"><i><font color="Green" title="E49">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">G</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">H</font>
;<br/>

<b>thus </b><a NAME="E7:83_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E3:83_1_1_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="grcat_1.html#E6:83_1_1_1_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="grcat_1.html#D27" target="_self" title="GRCAT_1:def.27">Def27</a></i>;<br/>


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

<b>thus </b><a NAME="E2:83_1_1_1"/>
(  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Maroon" title="c7">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c8">G</font>, <font color="Maroon" title="c9">H</font> being  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><a NAME="E1:83_1_1_1_2"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c7">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c10">M</font> =  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font>;<br/>
<a NAME="E2:83_1_1_1_2"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:83_1_1_1_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="grcat_1.html#D27" target="_self" title="GRCAT_1:def.27">Def27</a></i>;<br/>
<a NAME="E3:83_1_1_1_2"/>
 <a href="grcat_1.html#K18" title="GRCAT_1:func.18">Morphs</a> <font color="Maroon" title="c8">G</font>,<font color="Maroon" title="c9">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="grcat_1.html#K18" title="GRCAT_1:func.18">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="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:83_1_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E2:83_1_1_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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="83_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:83_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font>
 ;<br/><a NAME="E2:83_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Maroon" title="c7">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1: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="grcat_1.html#E5:83_1_1"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_2"/>
<font color="Maroon" title="c7">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a> <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">T'</font> = <font color="Maroon" title="c6">T</font> as   <a href="grcat_1.html#NM5" title="GRCAT_1:NM.5">GroupMorphism_DOMAIN</a> <b>by </b><i><a class="ref" href="grcat_1.html#D23" target="_self" title="GRCAT_1:def.23">Def23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">T'</font>
;<br/>

<b>thus </b><a NAME="E8:83_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="c7">T'</font> iff  ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1: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="grcat_1.html#E5:83_1_1"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>


</div>
