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

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">K</font>;<br/><b>let </b><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_1"/>
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> or <font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1"><b>suppose </b></a><a NAME="E1:141_1_1"/><i><font color="Green" title="E115">A1</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 ;<br/><div class="add"><a NAME="E2:141_1_1"/><b>then </b><i><font color="Green" title="E116">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/><b>thus </b><a NAME="E3:141_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:141_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E3:141_1_1_1"/><i><font color="Green" title="E117">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:141_1_1"><i><font color="Green" title="E116">A2</font></i></a>, <a class="ref" href="vectsp10.html#T4" title="VECTSP10:th.4">VECTSP10:4</a></i>;<br/>
<font color="Maroon" title="c1">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E3:141_1_1_1"><i><font color="Green" title="E117">A3</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp10.html#T2" title="VECTSP10:th.2">VECTSP10:2</a></i>
;<br/>
<b>hence </b><a NAME="E5:141_1_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font><b> such that </b><a NAME="E5:141_1_1"/><i><font color="Green" title="E117">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ;<br/><a NAME="E6:141_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:141_1_1"><i><font color="Green" title="E117">A3</font></i></a>, <a class="txt" href="matrix13.html#E1:141_1_1"><i><font color="Green" title="E115">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/><b>hence </b><a NAME="E7:141_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:141_1_1"><i><font color="Green" title="E116">A2</font></i></a>, <a class="ref" href="vectsp10.html#T4" title="VECTSP10:th.4">VECTSP10:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_2"><b>suppose </b></a><a NAME="E1:141_1_2"/><i><font color="Green" title="E115">A1</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:141_1_2"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:141_1_2_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">L</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><b> such that </b><br/><a NAME="E3:141_1_2_1"/><i><font color="Green" title="E116">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/>
<a NAME="E4:141_1_2_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:141_1_2"><i><font color="Green" title="E115">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:141_1_2_1"><i><font color="Green" title="E116">A2</font></i></a>, <a class="ref" href="vectsp_6.html#T44" title="VECTSP_6:th.44">VECTSP_6:44</a></i>;<br/>
<b>hence </b><a NAME="E5:141_1_2_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font><b> such that </b><a NAME="E4:141_1_2"/><i><font color="Green" title="E116">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>;<br/><b>consider </b><font color="Maroon" title="c8">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E6:141_1_2"/><i><font color="Green" title="E117">A4</font></i>: 
( <font color="Maroon" title="c8">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c8">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>and </b><br/><a NAME="E7:141_1_2"/><i><font color="Green" title="E118">A5</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">u</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font> holds <br/><font color="Maroon" title="c8">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S7" title="FUNCT_2:sch.7">FUNCT_2:sch 7</a>(<a class="txt" href="matrix13.html#E1:141_1_2"><i><font color="Green" title="E115">A1</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c9">L</font> = <font color="Maroon" title="c8">L</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">z</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font>;<br/><b>assume </b><a NAME="E1:141_1_2_2"/><i><font color="Green" title="E119">A6</font></i>: 
not <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><a NAME="E2:141_1_2_2"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">u</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:141_1_2_2"><i><font color="Green" title="E119">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:141_1_2_2"/>
<font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E7:141_1_2"><i><font color="Green" title="E118">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">L</font> = <font color="Maroon" title="c9">L</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4" title="VECTSP_6:def.4">VECTSP_6:def 4</a></i>;<br/><a NAME="E11:141_1_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c10">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:141_1_2_3"/><i><font color="Green" title="E119">A6</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c10">L</font>
 ;<br/>

<a NAME="E2:141_1_2_3"/>
<font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:141_1_2_3"><i><font color="Green" title="E119">A6</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/>
<a NAME="E3:141_1_2_3"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E7:141_1_2"><i><font color="Green" title="E118">A5</font></i></a>, <a class="txt" href="matrix13.html#E1:141_1_2_3"><i><font color="Green" title="E119">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:141_1_2_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">L</font> = <font color="Maroon" title="c10">L</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/><a NAME="E13:141_1_2"/>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c11">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:141_1_2"><i><font color="Green" title="E115">A1</font></i></a>, <a class="txt" href="matrix13.html#E4:141_1_2"><i><font color="Green" title="E116">A3</font></i></a>, <a class="txt" href="matrix13.html#E6:141_1_2"><i><font color="Green" title="E117">A4</font></i></a>, <a class="ref" href="vectsp_6.html#T44" title="VECTSP_6:th.44">VECTSP_6:44</a></i>;<br/><b>hence </b><a NAME="E14:141_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
