<?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">V</font>, <font color="Maroon" title="c3">W</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <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="c2">V</font>,<font color="Maroon" title="c3">W</font>;<br/>





<b>consider </b><font color="Maroon" title="c5">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A'</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="ranknull.html#T19" target="_self" title="RANKNULL:th.19">Th19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">B</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:60"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T17" title="VECTSP_9:th.17">VECTSP_9:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">B</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T24" title="VECTSP_9:th.24">VECTSP_9:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A'</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c8">B</font> <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">X</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E8:60"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">X</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E4:60"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c10">X</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">W</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c5">A</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">B</font> = <font color="Maroon" title="c8">B</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E12:60"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="ranknull.html#E4:60"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="ranknull.html#T22" target="_self" title="RANKNULL:th.22">Th22</a></i>;<br/>
<a NAME="E13:60"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">T</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="c2">V</font>
 
<b>by </b><i><a class="ref" href="ranknull.html#T7" target="_self" title="RANKNULL:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E2:60_1"/>
<font color="Maroon" title="c10">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:60"/><i><font color="Green" title="E49">A5</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<a NAME="E1:60_2"/>
<font color="Maroon" title="c10">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">X</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c10">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="ranknull.html#E12:60"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="ranknull.html#E13:60"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a></i>;<br/>
<a NAME="E2:60_2"/><b>then </b>
<font color="Maroon" title="c10">X</font>,<font color="Maroon" title="c11">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/>
<b>hence </b><a NAME="E3:60_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">X</font>
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:60"/><i><font color="Green" title="E50">A6</font></i>: 
<font color="Maroon" title="c11">C</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_3"/>
not <font color="Maroon" title="c11">C</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c11">C</font><b> such that </b><br/><a NAME="E3:60_3"/><i><font color="Green" title="E51">A7</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c14">l</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> 
 <b>and </b><br/><a NAME="E4:60_3"/><i><font color="Green" title="E52">A8</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c14">l</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="c3">W</font>
 <b>by </b><i><a class="ref" href="ranknull.html#T41" target="_self" title="RANKNULL:th.41">Th41</a></i>;<br/>
<a NAME="E5:60_3"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c10">X</font> st <font color="Maroon" title="c14">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c15">l'</font> = <font color="Maroon" title="c14">l</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c10">X</font> ;<br/>
<b>set </b><font color="Maroon" title="c16">m</font> = <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c15">l'</font>;<br/>
<b>take </b>
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c15">l'</font>
;<br/>

<b>thus </b><a NAME="E2:60_3_1"/>
<font color="Maroon" title="c14">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c15">l'</font></span>)</span>
 <b>by </b><i><a class="txt" href="ranknull.html#E12:60"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="ranknull.html#T43" target="_self" title="RANKNULL:th.43">Th43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c15">m</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A'</font><b> such that </b><br/><a NAME="E7:60_3"/><i><font color="Green" title="E53">A9</font></i>: 
<font color="Maroon" title="c14">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c15">m</font>
 ;<br/>
<a NAME="E8:60_3"/>
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">m</font></span>)</span> <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="c3">W</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E4:60"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="ranknull.html#E4:60_3"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="ranknull.html#E7:60_3"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="ranknull.html#T40" target="_self" title="RANKNULL:th.40">Th40</a></i>;<br/>
<a NAME="E9:60_3"/><b>then </b>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">m</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="ranknull.html#T10" target="_self" title="RANKNULL:th.10">Th10</a></i>;<br/>
<a NAME="E10:60_3"/><b>then </b>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">m</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> <font color="Maroon" title="c12">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E11:60_3"/><b>then </b>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">m</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> <font color="Maroon" title="c6">A'</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">n</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c6">A'</font><b> such that </b><br/><a NAME="E13:60_3"/><i><font color="Green" title="E54">A10</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">m</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="c16">n</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="E14:60_3"/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c16">n</font></span>)</span> <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">V</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E13:60_3"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</a></i>;<br/>
<a NAME="E15:60_3"/><b>then </b><i><font color="Green" title="E55">A11</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">n</font></span>)</span> <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">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T80" title="VECTSP_6:th.80">VECTSP_6:80</a></i>;<br/>
<a NAME="E16:60_3"/><i><font color="Green" title="E56">A12</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T74" title="VECTSP_6:th.74">VECTSP_6:74</a></i>;<br/>
<a NAME="E17:60_3"/><i><font color="Green" title="E57">A13</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A'</font>
 
<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="E18:60_3"/><i><font color="Green" title="E58">A14</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font>
 
<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="E19:60_3"/><i><font color="Green" title="E59">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A'</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E4:60"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E20:60_3"/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">A</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E17:60_3"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="ranknull.html#E18:60_3"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E21:60_3"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E16:60_3"><i><font color="Green" title="E56">A12</font></i></a>, <a class="txt" href="ranknull.html#E19:60_3"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">o</font> = <font color="Maroon" title="c15">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">n</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c13">B</font> <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="E23:60_3"/>
<font color="Maroon" title="c13">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E24:60_3"/><b>then </b><i><font color="Green" title="E60">A16</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ranknull.html#E15:60_3"><i><font color="Green" title="E55">A11</font></i></a>, <a class="ref" href="vectsp_7.html#D1" title="VECTSP_7:def.1">VECTSP_7:def 1</a></i>;<br/>
<a NAME="E25:60_3"/>
<font color="Maroon" title="c6">A'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E26:60_3"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E17:60_3"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="ranknull.html#E18:60_3"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="ranknull.html#T32" target="_self" title="RANKNULL:th.32">Th32</a>, <a class="ref" href="xboole_1.html#T64" title="XBOOLE_1:th.64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E27:60_3"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ranknull.html#E24:60_3"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E28:60_3"/><b>then </b>
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL: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="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></i>;<br/>
<b>hence </b><a NAME="E29:60_3"/>
contradiction
 <b>by </b><i><a class="txt" href="ranknull.html#E3:60_3"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="ranknull.html#E7:60_3"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="ranknull.html#T30" target="_self" title="RANKNULL:th.30">Th30</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c14">C</font> = <font color="Maroon" title="c11">C</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font></span>)</span> <b>by </b><i><a class="ref" href="ranknull.html#T12" target="_self" title="RANKNULL:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">L</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">C</font> as  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font> ;<br/>
<a NAME="E18:60"/>
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font></span>)</span> holds  <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font></span>)</span>;<br/>

<a NAME="E1:60_4"/><i><font color="Green" title="E51">A17</font></i>: 
<font color="Maroon" title="c16">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">v'</font> = <font color="Maroon" title="c16">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:60_4"/><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v'</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E1:60_4"><i><font color="Green" title="E51">A17</font></i></a>, <a class="ref" href="ranknull.html#T13" target="_self" title="RANKNULL:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">A'</font> = <font color="Maroon" title="c12">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="ranknull.html#T19" target="_self" title="RANKNULL:th.19">Th19</a></i>;<br/>
<a NAME="E6:60_4"/>
<font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c19">A'</font>, <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c19">A'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E4:60"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="ranknull.html#T33" target="_self" title="RANKNULL:th.33">Th33</a></i>;<br/>
<a NAME="E7:60_4"/><b>then </b><i><font color="Green" title="E53">A19</font></i>: 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c19">A'</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c19">A'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D4" title="VECTSP_5:def.4">VECTSP_5:def 4</a></i>;<br/>
<a NAME="E8:60_4"/>
<font color="Maroon" title="c18">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">u1</font>, <font color="Maroon" title="c21">u2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E10:60_4"/><i><font color="Green" title="E54">A20</font></i>: 
<font color="Maroon" title="c20">u1</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> <font color="Maroon" title="c19">A'</font>
 <b>and </b><br/><a NAME="E11:60_4"/><i><font color="Green" title="E55">A21</font></i>: 
<font color="Maroon" title="c21">u2</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">(<span class="default"><font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c19">A'</font></span>)</span>
 <b>and </b><br/><a NAME="E12:60_4"/><i><font color="Green" title="E56">A22</font></i>: 
<font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">u2</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E7:60_4"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/>
<a NAME="E13:60_4"/><i><font color="Green" title="E57">A23</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c20">u1</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="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">u2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E12:60_4"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="mod_2.html#D5" title="MOD_2:def.5">MOD_2:def 5</a></i>;<br/>
<a NAME="E14:60_4"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E15:60_4"/><b>then </b>
<font color="Maroon" title="c20">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E10:60_4"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<a NAME="E16:60_4"/><b>then </b>
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">u1</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="c3">W</font>
 
<b>by </b><i><a class="ref" href="ranknull.html#T10" target="_self" title="RANKNULL:th.10">Th10</a></i>;<br/>
<a NAME="E17:60_4"/><b>then </b><i><font color="Green" title="E58">A24</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">u2</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E13:60_4"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c19">A'</font><b> such that </b><br/><a NAME="E19:60_4"/><i><font color="Green" title="E59">A25</font></i>: 
<font color="Maroon" title="c21">u2</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="c22">l</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E11:60_4"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/>
<a NAME="E20:60_4"/><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c22">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL: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="c22">l</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ranknull.html#T29" target="_self" title="RANKNULL:th.29">Th29</a></i>;<br/>
<a NAME="E21:60_4"/><i><font color="Green" title="E61">A27</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c22">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c19">A'</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">C'</font> = <font color="Maroon" title="c14">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">W</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c22">l</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c23">C'</font> <b>by </b><i><a class="txt" href="ranknull.html#E20:60_4"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="ranknull.html#E21:60_4"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="vectsp_6.html#T25" title="VECTSP_6:th.25">VECTSP_6:25</a></i>;<br/>
<a NAME="E24:60_4"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c23">C'</font> st <font color="Maroon" title="c16">v</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="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c24">m</font>
;<br/>

<b>thus </b><a NAME="E1:60_4_1"/>
<font color="Maroon" title="c16">v</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="c24">m</font>
 <b>by </b><i><a class="txt" href="ranknull.html#E4:60"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="ranknull.html#E4:60_4"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="ranknull.html#E17:60_4"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="ranknull.html#E19:60_4"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="ranknull.html#T40" target="_self" title="RANKNULL:th.40">Th40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:60_4"/><b>then </b>
<font color="Maroon" title="c16">v</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> <font color="Maroon" title="c23">C'</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/>
<b>hence </b><a NAME="E26:60_4"/>
<font color="Maroon" title="c16">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">L</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:60"/><b>then </b><i><font color="Green" title="E51">A28</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T40" title="VECTSP_4:th.40">VECTSP_4:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">C</font> = <font color="Maroon" title="c14">C</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font></span>)</span> <b>by </b><i><a class="txt" href="ranknull.html#E15:60"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">C</font> = <font color="Maroon" title="c16">C</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c4">T</font> <b>by </b><i><a class="txt" href="ranknull.html#E19:60"><i><font color="Green" title="E51">A28</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E22:60"/><i><font color="Green" title="E52">A29</font></i>: 
 <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>;<br/>
<a NAME="E23:60"/><i><font color="Green" title="E53">A30</font></i>: 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">C</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>;<br/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">B</font>
<b>by </b><i><a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c4">T</font></span>)</span>
<b>by </b><i><a class="txt" href="ranknull.html#E8:60"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="ranknull.html#E14:60"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="ranknull.html#E22:60"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="ranknull.html#E23:60"><i><font color="Green" title="E53">A30</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E25:60"/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c4">T</font></span>)</span>
 ;<br/>


</div>
