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

<span class="kw">let </span><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">W</font> be   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>   <a href="zmodul03.html#V1" title="ZMODUL03:attr.1">finite-rank</a>  <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font> holds   <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K5" title="ZMODUL07:func.5">nullity</a> <font color="Olive" title="b1">T</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K5" title="ZMODUL07:func.5">nullity</a> <font color="Maroon" title="c3">T</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">A</font> =  the   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> =  the   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T29" onmouseover="rs('zmodul05/T29')" onmouseout="rh()">ZMODUL05:29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A9</font> = <font color="Maroon" title="c5">A9</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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 <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a>, <a class="ref" href="zmodul03.html#T15" onmouseover="rs('zmodul03/T15')" onmouseout="rh()">ZMODUL03:15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">B9</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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 <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><span class="kw"> such that </span><br/><a NAME="E4:47"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
( <font color="Maroon" title="c8">a</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> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> &amp; <font color="Maroon" title="c6">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B9</font> &amp; <font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T5" target="_self" onmouseover="rs('zmodul07/T5')" onmouseout="rh()">LmRankSX11</a></span>;<br/>
<a NAME="E5:47"/><span class="lab"><font color="Green" title="E25">X1</font></span>: 
<font color="Maroon" title="c1">V</font> is   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>   <a href="zmodul03.html#V1" title="ZMODUL03:attr.1">finite-rank</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T32" onmouseover="rs('zmodul01/T32')" onmouseout="rh()">ZMODUL01:32</a></span>;<br/>
<a NAME="E6:47"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="zmodul05.html#T2" onmouseover="rs('zmodul05/T2')" onmouseout="rh()">ZMODUL05:2</a></span>;<br/>
<a NAME="E7:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">X2</font></span>: 
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E25">X1</font></span></a>, <a class="ref" href="zmodul06.html#T52" onmouseover="rs('zmodul06/T52')" onmouseout="rh()">ZMODUL06:52</a></span>;<br/>
<a NAME="E8:47"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T2" onmouseover="rs('zmodul05/T2')" onmouseout="rh()">ZMODUL05:2</a></span>;<br/>
<a NAME="E9:47"/><span class="kw">then </span>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E26">X2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E10:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">X0</font></span>: 
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T3" onmouseover="rs('zmodul05/T3')" onmouseout="rh()">ZMODUL05:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c7">B9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A9</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="c7">B9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<span class="kw">reconsider </span><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="c1">V</font> ;<br/>
<a NAME="E13:47"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c7">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c3">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="c2">W</font> ;<br/>
<a NAME="E15:47"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c3">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E27">X0</font></span></a>, <a class="ref" href="zmodul07.html#T20" target="_self" onmouseover="rs('zmodul07/T20')" onmouseout="rh()">ZM05Th35</a></span>;<br/>
<a NAME="E16:47"/>
<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon" title="c11">C</font> is  <a href="vectsp_7.html#NV2" title="VECTSP_7:NV.2">linearly-dependent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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><span class="kw"> such that </span><br/><a NAME="E3:47_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">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> 
 <span class="kw">and </span><br/><a NAME="E4:47_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c12">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="c2">W</font>
 ;<br/>
<a NAME="E5:47_1"/>
 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="c12">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K5" title="ZMODUL05:func.5">@*</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c13">l9</font> = <font color="Maroon" title="c12">l</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c10">X</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">m</font> = <font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K6" title="ZMODUL05:func.6">#</a> <font color="Maroon" title="c13">l9</font>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K6" title="ZMODUL05:func.6">#</a> <font color="Maroon" title="c13">l9</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K5" title="ZMODUL05:func.5">@*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K6" title="ZMODUL05:func.6">#</a> <font color="Maroon" title="c13">l9</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E2:47_1_1"/>
<font color="Maroon" title="c12">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K5" title="ZMODUL05:func.5">@*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K6" title="ZMODUL05:func.6">#</a> <font color="Maroon" title="c13">l9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="zmodul05.html#T60" onmouseover="rs('zmodul05/T60')" onmouseout="rh()">ZMODUL05:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">m</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c7">B9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A9</font><span class="kw"> such that </span><br/><a NAME="E7:47_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c12">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="zmodul05.html#K5" title="ZMODUL05:func.5">@*</a> <font color="Maroon" title="c13">m</font>
 ;<br/>
<a NAME="E8:47_1"/>
<font color="Maroon" title="c3">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="c13">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="c2">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E7:47_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E27">X0</font></span></a>, <a class="ref" href="zmodul07.html#T21" target="_self" onmouseover="rs('zmodul07/T21')" onmouseout="rh()">ZM05Th59</a></span>;<br/>
<a NAME="E9:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">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="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T20" onmouseover="rs('zmodul05/T20')" onmouseout="rh()">ZMODUL05:20</a></span>;<br/>
<a NAME="E10:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">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>  the   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a></span>;<br/>
<a NAME="E11:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">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">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul03.html#T20" onmouseover="rs('zmodul03/T20')" onmouseout="rh()">ZMODUL03:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c6">A9</font><span class="kw"> such that </span><br/><a NAME="E13:47_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">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="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T64" onmouseover="rs('zmodul02/T64')" onmouseout="rh()">ZMODUL02:64</a></span>;<br/>
<a NAME="E14:47_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c14">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="c13">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="c14">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">B9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A9</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="zmodul02.html#T40" onmouseover="rs('zmodul02/T40')" onmouseout="rh()">ZMODUL02:40</a>, <a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a></span>;<br/>
<a NAME="E15:47_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A9</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">VECTSP_6:def 4</a></span>;<br/>
<a NAME="E16:47_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">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="c14">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="c7">B9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A9</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>;<br/>
<a NAME="E17:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">o</font> = <font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c14">n</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c7">B9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">VECTSP_6:def 4</a></span>;<br/>
<a NAME="E19:47_1"/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c14">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="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="vectsp_1.html#T19" onmouseover="rs('vectsp_1/T19')" onmouseout="rh()">VECTSP_1:19</a></span>;<br/>
<a NAME="E20:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c14">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="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T55" onmouseover="rs('zmodul02/T55')" onmouseout="rh()">ZMODUL02:55</a></span>;<br/>
<a NAME="E21:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A12</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D1" onmouseover="rs('vectsp_7/D1')" onmouseout="rh()">VECTSP_7:def 1</a></span>;<br/>
<a NAME="E22:47_1"/>
<font color="Maroon" title="c6">A9</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">B9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/>
<a NAME="E23:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c14">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="c13">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="c14">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:47_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="xboole_1.html#T64" onmouseover="rs('xboole_1/T64')" onmouseout="rh()">XBOOLE_1:64</a>, <a class="ref" href="zmodul05.html#T48" onmouseover="rs('zmodul05/T48')" onmouseout="rh()">ZMODUL05:48</a></span>;<br/>
<a NAME="E24:47_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:47_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/>
<a NAME="E25:47_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c13">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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E26:47_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E7:47_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a>, <a class="ref" href="zmodul05.html#D8" onmouseover="rs('zmodul05/D8')" onmouseout="rh()">ZMODUL05:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">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="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 <font color="Maroon" title="c2">W</font> ;<br/>
<a NAME="E18:47"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">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="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="zmodul07.html#E2"><span class="lab"><font color="Green" title="E2">LmDOMRNG</font></span></a></span>;<br/>
<a NAME="E19:47"/><span class="kw">then </span>
<font color="Maroon" title="c10">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E20:47"/><span class="kw">then </span>
<font color="Maroon" title="c10">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">X</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c10">X</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="card_1.html#T33" onmouseover="rs('card_1/T33')" onmouseout="rh()">CARD_1:33</a></span>;<br/>
<a NAME="E21:47"/><span class="kw">then </span>
<font color="Maroon" title="c10">X</font>,<font color="Maroon" title="c12">C</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T129" onmouseover="rs('relat_1/T129')" onmouseout="rh()">RELAT_1:129</a></span>;<br/>
<a NAME="E22:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">aT</font> = <font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font></span>)</span> as   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c2">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T42" onmouseover="rs('zmodul01/T42')" onmouseout="rh()">ZMODUL01:42</a></span>;<br/>
<a NAME="E24:47"/><span class="lab"><font color="Green" title="E31">L1</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c7">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c10">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="zmodul07.html#T23" target="_self" onmouseover="rs('zmodul07/T23')" onmouseout="rh()">LMTh44</a></span>;<br/>
<a NAME="E25:47"/>
 for <font color="Olive" title="b1">v</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <font color="Maroon" title="c2">W</font>  st <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">aT</font> holds <br/><font color="Olive" title="b1">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="c12">C</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">v</font> be   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <font color="Maroon" title="c2">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">aT</font> implies <font color="Maroon" title="c14">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="c12">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_2"/>
<font color="Maroon" title="c14">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">aT</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">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="c12">C</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">tw</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:47_2"/><span class="lab"><font color="Green" title="E32">X1</font></span>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c15">tw</font>
 ;<br/>
<a NAME="E4:47_2"/>
( <font color="Maroon" title="c15">tw</font> is   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c15">tw</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="c3">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T25" onmouseover="rs('zmodul01/T25')" onmouseout="rh()">ZMODUL01:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">w</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E6:47_2"/><span class="lab"><font color="Green" title="E33">X2</font></span>: 
<font color="Maroon" title="c15">tw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T23" onmouseover="rs('zmodul05/T23')" onmouseout="rh()">ZMODUL05:23</a></span>;<br/>
<span class="lab"><font color="Green" title="E34">X3</font></span>: <font color="Maroon" title="c14">v</font> = 
<font color="Maroon" title="c8">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_2"><span class="lab"><font color="Green" title="E32">X1</font></span></a>, <a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E33">X2</font></span></a>, <a class="ref" href="zmodul01.html#T29" onmouseover="rs('zmodul01/T29')" onmouseout="rh()">ZMODUL01:29</a></span>
<br/>.= 
<font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c16">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D2" onmouseover="rs('mod_2/D2')" onmouseout="rh()">MOD_2:def 2</a></span>
;<br/>
<a NAME="E8:47_2"/>
<font color="Maroon" title="c8">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c16">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K1" title="ZMODUL02:func.1">*</a> <font color="Maroon" title="c1">V</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">aw</font> = <font color="Maroon" title="c8">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c16">w</font> as   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<a NAME="E10:47_2"/>
<font color="Maroon" title="c17">aw</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font>
 
;<br/>
<a NAME="E11:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">aw</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="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="zmodul01.html#T24" onmouseover="rs('zmodul01/T24')" onmouseout="rh()">ZMODUL01:24</a></span>;<br/>
<a NAME="E12:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">aw</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:47_2"/>
<font color="Maroon" title="c14">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="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:47"><span class="lab"><font color="Green" title="E31">L1</font></span></a>, <a class="txt" href="#E7:47_2"><span class="lab"><font color="Green" title="E34">X3</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>, <a class="ref" href="zmodul06.html#T40" onmouseover="rs('zmodul06/T40')" onmouseout="rh()">ZMODUL06:40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:47"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font></span>)</span> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T44" onmouseover="rs('zmodul01/T44')" onmouseout="rh()">ZMODUL01:44</a></span>;<br/>
<a NAME="E27:47"/><span class="kw">then </span>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T2" onmouseover="rs('zmodul05/T2')" onmouseout="rh()">ZMODUL05:2</a></span>;<br/>
<a NAME="E28:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">X4</font></span>: 
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="zmodul06.html#T52" onmouseover="rs('zmodul06/T52')" onmouseout="rh()">ZMODUL06:52</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">CC</font> = <font color="Maroon" title="c12">C</font> as   <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="c3">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T22" onmouseover="rs('zmodul05/T22')" onmouseout="rh()">ZMODUL05:22</a></span>;<br/>
<a NAME="E30:47"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">CC</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font>
 
;<br/>
<a NAME="E31:47"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">C</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul03.html#T20" onmouseover="rs('zmodul03/T20')" onmouseout="rh()">ZMODUL03:20</a></span>;<br/>
<a NAME="E32:47"/><span class="kw">then </span>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c3">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T2" onmouseover="rs('zmodul05/T2')" onmouseout="rh()">ZMODUL05:2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E33">X5</font></span>:  <a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font> = 
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c12">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:47"><span class="lab"><font color="Green" title="E32">X4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T3" onmouseover="rs('zmodul05/T3')" onmouseout="rh()">ZMODUL05:3</a></span>
;<br/>
<a NAME="E34:47"/>
 <a href="zmodul07.html#K5" title="ZMODUL07:func.5">nullity</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a>  the   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul03.html#D5" onmouseover="rs('zmodul03/D5')" onmouseout="rh()">ZMODUL03:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:47"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K5" title="ZMODUL07:func.5">nullity</a> <font color="Maroon" title="c3">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E22:47"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E27">X0</font></span></a>, <a class="txt" href="#E33:47"><span class="lab"><font color="Green" title="E33">X5</font></span></a>, <a class="ref" href="card_2.html#T40" onmouseover="rs('card_2/T40')" onmouseout="rh()">CARD_2:40</a>, <a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
