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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font>, <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="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c4">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c2">W</font>;<br/><b>let </b><font color="Maroon" title="c5">l</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">V</font>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c6">v</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="c5">l</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font><b> such that </b><br/><a NAME="E4:51"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 <b>and </b><br/><a NAME="E5:51"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E1:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="ranknull.html#E2:51"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="ranknull.html#T34" target="_self" title="RANKNULL:th.34">Th34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:51_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">v</font>

<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="ranknull.html#E2:51_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
;<br/><a NAME="E4:51_1_1"/><b>then </b>
 <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E5:51"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="ranknull.html#E1:51_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="rlvect_2.html#T15" title="RLVECT_2:th.15">RLVECT_2:15</a></i>;<br/><b>hence </b><a NAME="E5:51_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="ref" href="ranknull.html#D5" target="_self" title="RANKNULL:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><a NAME="E1:51_1_2"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:51_1_2"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="ranknull.html#E5:51"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>;<br/><a NAME="E3:51_1_2"/><i><font color="Green" title="E43">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/><i><font color="Green" title="E44">A10</font></i>: <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">v</font>

<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="ranknull.html#E3:51_1_2"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
;<br/><a NAME="E5:51_1_2"/><i><font color="Green" title="E45">A11</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font> <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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_2_2"/><i><font color="Green" title="E46">A12</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><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="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:51_1_2_2_1"/><i><font color="Green" title="E47">A13</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><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="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:51_1_2_2_1"/><i><font color="Green" title="E48">A14</font></i>: 
<font color="Maroon" title="c8">x</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="c1">F</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E1:51_1_2_2_1"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:51_1_2_2_1"/><i><font color="Green" title="E49">A15</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E1:51_1_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<div><i><font color="Green" title="E50">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_2_2_1_1"/>
<font color="Maroon" title="c9">y</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="c5">l</font>
 ;<br/><a NAME="E2:51_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E4:51_1_2_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:51_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ranknull.html#E4:51"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font> <b>by </b><i><a class="txt" href="ranknull.html#E4:51_1_2_2_1"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/>
<a NAME="E7:51_1_2_2_1"/>
<font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E2:51_1_2_2_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="ranknull.html#E5:51_1_2_2_1"><i><font color="Green" title="E50">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:51_1_2_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E3:51_1_2"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="ranknull.html#E4:51_1_2_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:51_1_2_2"/>
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><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="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</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="E1:51_1_2_2_2"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_1_2_2_2"/><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">l</font>
 <b>and </b><br/><a NAME="E4:51_1_2_2_2"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>and </b><br/><a NAME="E5:51_1_2_2_2"/><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E1:51_1_2_2_2"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<div><i><font color="Green" title="E51">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_2_2_2_1"/>
<font color="Maroon" title="c9">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="c5">l</font>
 ;<br/><a NAME="E2:51_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E4:51_1_2_2_2"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:51_1_2_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ranknull.html#E4:51"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font> <b>by </b><i><a class="txt" href="ranknull.html#E3:51_1_2_2_2"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/>
<a NAME="E8:51_1_2_2_2"/>
<font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</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="c1">F</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E6:51_1_2_2_2"><i><font color="Green" title="E51">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:51_1_2_2_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><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="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="ranknull.html#E5:51_1_2_2_2"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:51_1_2_2"/>
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font> <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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="ranknull.html#E1:51_1_2_2"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:51_1_2"/><i><font color="Green" title="E46">A22</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_2_3"/><i><font color="Green" title="E47">A23</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/>
<i><font color="Green" title="E48">A24</font></i>: <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">v</font>

<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="ranknull.html#E1:51_1_2_3"><i><font color="Green" title="E47">A23</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
;<br/>
<b>assume </b><a NAME="E3:51_1_2_3"/>
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:51_1_2_3"/><i><font color="Green" title="E49">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font>
 <b>and </b><br/><a NAME="E6:51_1_2_3"/><i><font color="Green" title="E50">A26</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:51_1_2_3"/><i><font color="Green" title="E51">A27</font></i>: 
<font color="Maroon" title="c8">x</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="c1">F</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E5:51_1_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="ranknull.html#E5:51_1_2_3"><i><font color="Green" title="E49">A25</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:51_1_2_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E2:51_1_2_3"><i><font color="Green" title="E48">A24</font></i></a>, <a class="txt" href="ranknull.html#E6:51_1_2_3"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:51_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="ranknull.html#E2:51"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="ranknull.html#E7:51_1_2_3"><i><font color="Green" title="E51">A27</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/>


</div><b>end;</b></div> <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ranknull.html#E2:51_1_2"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="ranknull.html#E6:51_1_2"><i><font color="Green" title="E46">A22</font></i></a>, <a class="ref" href="rlvect_2.html#T18" title="RLVECT_2:th.18">RLVECT_2:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ranknull.html#E4:51_1_2"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="ranknull.html#E5:51_1_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="rlvect_2.html#T15" title="RLVECT_2:th.15">RLVECT_2:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</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="c1">F</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T15" title="RLVECT_2:th.15">RLVECT_2:15</a></i>
<br/>.= 
<font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E8:51_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="ref" href="ranknull.html#D5" target="_self" title="RANKNULL:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
