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

<b>set </b><font color="Maroon" title="c3">D</font> = <span class="p1">{<span class="default"> <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) where <font color="Olive" title="b1">f</font> is    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> : <font color="Olive" title="b1">f</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c4">F</font> being    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font><b> such that </b><br/><a NAME="E3:50_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c5">f</font> #)
 <b>and </b><br/><a NAME="E4:50_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:50"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="ringcat1.html#E22"><i><font color="Green" title="E14">Lm7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">f0</font> = <font color="Maroon" title="c5">f</font> as    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E6:50_1_1"/>
 <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c6">f0</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) where <font color="Olive" title="b1">f</font> is    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> : <font color="Olive" title="b1">f</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ringcat1.html#E4:50_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">D</font> = <span class="p1">{<span class="default"> <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) where <font color="Olive" title="b1">f</font> is    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> : <font color="Olive" title="b1">f</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> </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>  ;<br/>
<a NAME="E8:50_1_1"/><i><font color="Green" title="E32">A3</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> <font color="Maroon" title="c7">D</font> holds <br/><font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 ;<br/>

<a NAME="E2:50_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being   <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) &amp; <font color="Olive" title="b1">f</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> )
 
;<br/>
<b>hence </b><a NAME="E3:50_1_1_1"/>
<font color="Maroon" title="c8">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E17"><i><font color="Green" title="E10">Lm5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:50_1_1"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">D</font> holds  <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 
;<br/>
<a NAME="E10:50_1_1"/><i><font color="Green" title="E34">A5</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> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> 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">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_2"/>
<font color="Maroon" title="c8">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> ;<br/>
<a NAME="E3:50_1_1_2"/><i><font color="Green" title="E35">A6</font></i>: 
(  <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:50"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self" title="RINGCAT1:def.9">Def9</a></i>;<br/>
<a NAME="E4:50_1_1_2"/><i><font color="Green" title="E36">A7</font></i>: 
the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c9">x</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E15"><i><font color="Green" title="E8">Lm3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = the <a href="ringcat1.html#U3" title="RINGCAT1:sel.3">Fun</a> of <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="ringcat1.html#E3:50_1_1_2"><i><font color="Green" title="E35">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">f</font> as    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E7:50_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#G1" title="RINGCAT1:aggr.1">RingMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c11">g</font> #)
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:50_1_1_2"><i><font color="Green" title="E35">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:50_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E4:50_1_1_2"><i><font color="Green" title="E36">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">D</font> as    <a href="ringcat1.html#M4" title="RINGCAT1:mode.4">RingMorphism_DOMAIN</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="ringcat1.html#E9:50_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="ringcat1.html#T18" target="_self" title="RINGCAT1:th.18">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">D</font>
;<br/>

<b>thus </b><a NAME="E12:50_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="c8">D</font> iff <font color="Olive" title="b1">x</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E8:50_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="ringcat1.html#E10:50_1_1"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>


</div>
