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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1"/>
( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_1"/>
 <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_2"/>
 not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">X</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c3">V</font> =  <a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font>;<br/><span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c2">X</font>;<br/><a NAME="E3:42_1_2"/>
 for <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c2">X</font>  st  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/> <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">l</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c2">X</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="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</span> implies  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_2_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</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">  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> =  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">l</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">C</font> =  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>;<br/>
<a NAME="E3:42_1_2_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K6" title="RANKNULL:func.6">!</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c5">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T24" onmouseover="rs('ranknull/T24')" onmouseout="rh()">RANKNULL:24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:42_1_2_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">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> 
 ; <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="c8">f</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:42_1_2_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T4" onmouseover="rs('subset_1/T4')" onmouseout="rh()">SUBSET_1:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</span> ;<br/>
<a NAME="E9:42_1_2_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K6" title="RANKNULL:func.6">!</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">m</font> = <font color="Maroon" title="c5">l</font> <a href="ranknull.html#K6" title="RANKNULL:func.6">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">l</font> = <font color="Maroon" title="c5">l</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_2_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> =  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c15">u</font> =  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c11">n</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<a NAME="E14:42_1_2_1"/>
<font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:42_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
(  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c11">n</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T25" onmouseover="rs('ranknull/T25')" onmouseout="rh()">RANKNULL:25</a>, <a class="ref" href="vectsp_6.html#T17" onmouseover="rs('vectsp_6/T17')" onmouseout="rh()">VECTSP_6:17</a></span>;<br/>
<a NAME="E16:42_1_2_1"/>
<font color="Maroon" title="c13">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">g</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="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/>
<a NAME="E17:42_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#T5" target="_self" onmouseover="rs('bspace/T5')" onmouseout="rh()">Th5</a>, <a class="ref" href="bspace.html#T6" target="_self" onmouseover="rs('bspace/T6')" onmouseout="rh()">Th6</a>, <a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E18:42_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1_2_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<a NAME="E19:42_1_2_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c2">X</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="E20:42_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_2_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="bspace.html#T30" target="_self" onmouseover="rs('bspace/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E22:42_1_2_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T65" onmouseover="rs('card_1/T65')" onmouseout="rh()">CARD_1:65</a></span>;<br/>
<a NAME="E23:42_1_2_1"/>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:42_1_2_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E24:42_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#D3" target="_self" onmouseover="rs('bspace/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E25:42_1_2_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> holds <br/><font color="Olive" title="b1">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">g</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</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="c17">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c17">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> implies <font color="Maroon" title="c17">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1_2_1_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E2:42_1_2_1_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</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="c17">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span><br/>

<a NAME="E3:42_1_2_1_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c2">X</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="E4:42_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">g</font> is 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="bspace.html#T30" target="_self" onmouseover="rs('bspace/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E6:42_1_2_1_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T65" onmouseover="rs('card_1/T65')" onmouseout="rh()">CARD_1:65</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_1_1"><span class="kw">now </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">  not <font color="Maroon" title="c17">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_1_2_1_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</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/><a NAME="E2:42_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_2_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="bspace.html#D3" target="_self" onmouseover="rs('bspace/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:42_1_2_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E6:42_1_2_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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">hence </span><a NAME="E8:42_1_2_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:42_1_2_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c17">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:42_1_2_1_2"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
( <font color="Maroon" title="c17">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c17">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">and </span><br/><a NAME="E3:42_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c17">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D6" onmouseover="rs('vectsp_6/D6')" onmouseout="rh()">VECTSP_6:def 6</a></span>;<br/>
<a NAME="E4:42_1_2_1_2"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c17">F</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#T34" target="_self" onmouseover="rs('bspace/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E5:42_1_2_1_2"/>
 for <font color="Olive" title="b1">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">F</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c2">X</font></span>)</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"> ( <font color="Maroon" title="c18">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1_2_1_2_1"/>
<font color="Maroon" title="c18">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E2:42_1_2_1_2_1"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">m</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c19">R</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">L</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font>;<br/>
<a NAME="E3:42_1_2_1_2_1"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D5" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">VECTSP_6:def 5</a></span>;<br/>
<a NAME="E4:42_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#D9" target="_self" onmouseover="rs('bspace/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E5:42_1_2_1_2_1"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">k</font> &amp; <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">k</font> &amp; <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span><br/>

<a NAME="E2:42_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
<font color="Maroon" title="c21">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a></span>;<br/>
<a NAME="E3:42_1_2_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D5" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">VECTSP_6:def 5</a></span>;<br/>
<a NAME="E4:42_1_2_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:42_1_2_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a></span>;<br/>
<a NAME="E6:42_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D5" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">VECTSP_6:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">Fk</font> = <font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">k</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<a NAME="E8:42_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">m</font> <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="c5">l</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="E9:42_1_2_1_2_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:42_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c21">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a></span>;<br/>
<a NAME="E11:42_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E12:42_1_2_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="bspace.html#T35" target="_self" onmouseover="rs('bspace/T35')" onmouseout="rh()">Th35</a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E13:42_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">Fk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a></span>;<br/>
<a NAME="E14:42_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E10:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E11:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E15:42_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<font color="Maroon" title="c22">Fk</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E16:42_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
<font color="Maroon" title="c22">Fk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c22">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</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"> contradiction</span><br/>

<a NAME="E2:42_1_2_1_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E8:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_1_2_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E4:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="ref" href="bspace.html#D9" target="_self" onmouseover="rs('bspace/D9')" onmouseout="rh()">Def9</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:42_1_2_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E13:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E16:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E15:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:42_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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="E6:42_1_2_1_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<a NAME="E7:42_1_2_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:42_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E47">A20</font></span></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="E6:42_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="bspace.html#K11" title="BSPACE:func.11">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1_2"><span class="lab"><font color="Green" title="E41">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:42_1_2_1_2"/>
<font color="Maroon" title="c14">t</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_2_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E4:42_1_2_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="bspace.html#T5" target="_self" onmouseover="rs('bspace/T5')" onmouseout="rh()">Th5</a>, <a class="ref" href="matrix_3.html#T11" onmouseover="rs('matrix_3/T11')" onmouseout="rh()">MATRIX_3:11</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="E27:42_1_2_1"/>
<font color="Maroon" title="c13">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c11">n</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1_2_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="ranknull.html#T27" onmouseover="rs('ranknull/T27')" onmouseout="rh()">RANKNULL:27</a></span>;<br/>
<a NAME="E28:42_1_2_1"/><span class="kw">then </span>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1: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="c11">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T44" onmouseover="rs('vectsp_6/T44')" onmouseout="rh()">VECTSP_6:44</a></span>;<br/>
<a NAME="E29:42_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c15">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#D5" target="_self" onmouseover="rs('bspace/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E30:42_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</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="c15">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#T15" target="_self" onmouseover="rs('bspace/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E31:42_1_2_1"/>
<font color="Maroon" title="c6">s</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="bspace.html#D3" target="_self" onmouseover="rs('bspace/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:42_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_1_2_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E30:42_1_2_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="txt" href="#E26:42_1_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E18:42_1_2_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</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">hence </span><a NAME="E4:42_1_2"/>
 <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</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>; <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></div><span class="kw">end;</span></div>

</div>
