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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> ;<br/>
<a NAME="E1:111_1_1"/>
for <font color="Olive">y</font>, <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span>  holds <br/><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:111_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:111_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 ;<br/>
<a NAME="E4:111_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 
<b>by </b><i><a class="ref" href="ideal_1.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E5:111_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E6:111_1_1_1"/>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:111_1_1"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font> is <a href="ideal_1.html#V2" target="_self">left-ideal</a>
 <b>by </b><i/>;<br/>


</div>
