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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L2">RLSStruct</a> , <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">x2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>, <font color="Maroon">y2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</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="rlvect_1.html#NM1">VECTOR</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>assume </b><a NAME="E1:6"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/>

<a NAME="E2:6"/><i><font color="Green">E38</font></i>: 
<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">(<span class="default"><a href="convfun1.html#K1" target="_self">Add_in_Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">u2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>;<br/>
<a NAME="E3:6"/><i><font color="Green">E41</font></i>: 
<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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u1</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">u2</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E3:6_1"/>
<span class="p1">(<span class="default"><a href="convfun1.html#K1" target="_self">Add_in_Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:6_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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6"/>
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:6"/>
<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">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>


</div>
