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

<b>set </b><font color="Maroon">H</font> = <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font>;<br/>
<b>thus </b><a NAME="E1:52_1_1"/>
<font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font> is    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D10" target="_self">IDEAL_1:def 10</a><br/>

<b>assume </b><a NAME="E1:52_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<a NAME="E2:52_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:52_1_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:52_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:52_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E7:52_1_1_1_1_1"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> st <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_2"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:52_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:52_1_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:52_1_1_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:52_1_1_1_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E8:52_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:52_1_1_1_1_2"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> st <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
