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

<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c4">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">T</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> ;<br/>
<a NAME="E2:35"/><i><font color="Green" title="E24">A2</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">u</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:35_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font>
 <b>and </b><br/><a NAME="E4:35_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 ;<br/>
<a NAME="E5:35_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D12" target="_self" title="VECTSP_6:def.12">Def12</a></i>;<br/>
<a NAME="E6:35_1_1"/><b>then </b>
<font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E4:35_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/>
<b>hence </b><a NAME="E7:35_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:35_1_1"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:35_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">v</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font>
 <b>and </b><br/><a NAME="E5:35_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 ;<br/>
<a NAME="E6:35_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D12" target="_self" title="VECTSP_6:def.12">Def12</a></i>;<br/>
<a NAME="E7:35_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="vectsp_6.html#E5:35_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="vectsp_6.html#E4:35_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:35"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c4">L</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>


</div>
