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<a NAME="E1:7_1_1"/>
for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> holds  <font color="Olive">u1</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u2</font> <a href="hidden.html#R1">=</a> <font color="Olive">u2</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:7_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E3:7_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:7_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E8:7_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E11:7_1_1_1"/>
<font color="Maroon">u1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E12:7_1_1_1"/>
<font color="Maroon">u1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u1</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_1"/>
 <a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a></i>;<br/>


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