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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_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="47_1_1"><b>suppose </b></a><a NAME="E1:47_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:47_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E3:47_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>
 <b>by </b><i><a class="txt" href="bspace.html#E1:47_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="bspace.html#T36" target="_self" title="BSPACE:th.36">Th36</a>, <a class="ref" href="vectsp_7.html#T4" title="VECTSP_7:th.4">VECTSP_7:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><a NAME="E1:47_1_2"/>
not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><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#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>set </b><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/><b>set </b><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="E3:47_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" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E1:47_1_2_1"/><i><font color="Green" title="E34">A2</font></i>: 
 <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>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">C</font> =  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">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/>
<b>assume </b><a NAME="E3:47_1_2_1"/><i><font color="Green" title="E35">A3</font></i>: 
 <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> 
 ;<br/>

<b>consider </b><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><b> such that </b><br/><a NAME="E5:47_1_2_1"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 <b>by </b><i><a class="txt" href="bspace.html#E3:47_1_2_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<b>reconsider </b><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/>
<a NAME="E7:47_1_2_1"/><i><font color="Green" title="E37">A5</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E8:47_1_2_1"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c9">f</font> is <a href="bspace.html#V1" title="BSPACE:attr.1">Singleton</a>
 
<b>by </b><i><a class="txt" href="bspace.html#E5:47_1_2_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="bspace.html#E7:47_1_2_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="bspace.html#T30" target="_self" title="BSPACE:th.30">Th30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:47_1_2_1"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E11:47_1_2_1"/><i><font color="Green" title="E40">A8</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="bspace.html#E8:47_1_2_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="bspace.html#D9" target="_self" title="BSPACE:def.9">Def9</a></i>;<br/>
<a NAME="E12:47_1_2_1"/><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E11:47_1_2_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:47_1_2_1"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E12:47_1_2_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="bspace.html#E10:47_1_2_1"><i><font color="Green" title="E39">A7</font></i></a></i>;<br/>
<a NAME="E15:47_1_2_1"/><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">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> 
 
<b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/>
<a NAME="E16:47_1_2_1"/><i><font color="Green" title="E44">A12</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font> holds <br/><font color="Olive" title="b1">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">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" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">g</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2_1_1"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font>
 <b>and </b><br/><a NAME="E2:47_1_2_1_1"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 ;<br/>

<a NAME="E3:47_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>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E4:47_1_2_1_1"/><b>then </b><i><font color="Green" title="E47">A15</font></i>: 
<font color="Maroon" title="c12">g</font> is <a href="bspace.html#V1" title="BSPACE:attr.1">Singleton</a>
 
<b>by </b><i><a class="txt" href="bspace.html#E2:47_1_2_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="bspace.html#T30" target="_self" title="BSPACE:th.30">Th30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:47_1_2_1_1"/><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E7:47_1_2_1_1"/><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="bspace.html#E4:47_1_2_1_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="bspace.html#D9" target="_self" title="BSPACE:def.9">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="bspace.html#E6:47_1_2_1_1"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2_1_1_1"/>
<font color="Maroon" title="c12">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 ;<br/><a NAME="E2:47_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="bspace.html#E7:47_1_2_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E3:47_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bspace.html#E11:47_1_2_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="bspace.html#E1:47_1_2_1_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="bspace.html#E7:47_1_2_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:47_1_2_1_1"/>
<font color="Maroon" title="c12">g</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">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> 
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c12">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/>
<b>reconsider </b><font color="Maroon" title="c13">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="c12">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="c12">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">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="c12">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="c12">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">t</font> =  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">m</font>, <font color="Maroon" title="c16">u</font> =  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c14">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="E21:47_1_2_1"/><i><font color="Green" title="E45">A18</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="ranknull.html#T24" title="RANKNULL:th.24">RANKNULL:24</a></i>;<br/>
<a NAME="E22:47_1_2_1"/><i><font color="Green" title="E46">A19</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">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>
 
<b>by </b><i><a class="txt" href="bspace.html#E5:47_1_2_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">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> <b>by </b><i><a class="txt" href="bspace.html#E21:47_1_2_1"><i><font color="Green" title="E45">A18</font></i></a></i>;<br/>
<a NAME="E24:47_1_2_1"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c14">n</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E22:47_1_2_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="ranknull.html#T27" title="RANKNULL:th.27">RANKNULL:27</a></i>;<br/>
<a NAME="E25:47_1_2_1"/><i><font color="Green" title="E48">A21</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c17">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="c13">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="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bspace.html#E24:47_1_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="vectsp_6.html#T77" title="VECTSP_6:th.77">VECTSP_6:77</a></i>;<br/>
<a NAME="E26:47_1_2_1"/><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c16">u</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E25:47_1_2_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="bspace.html#D5" target="_self" title="BSPACE:def.5">Def5</a></i>;<br/>
<a NAME="E27:47_1_2_1"/><i><font color="Green" title="E50">A23</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">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="c16">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bspace.html#E26:47_1_2_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="ref" href="bspace.html#T15" target="_self" title="BSPACE:th.15">Th15</a></i>;<br/>
<a NAME="E28:47_1_2_1"/><i><font color="Green" title="E51">A24</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_2_1_2"/><i><font color="Green" title="E52">A25</font></i>: 
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="relat_1.html#K2" title="RELAT_1:func.2">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="c13">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c11">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#K2" title="FINSEQ_2:func.2">|-&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" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_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>
 <b>and </b><br/><a NAME="E2:47_1_2_1_2_1"/><i><font color="Green" title="E53">A26</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">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="c13">m</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c19">L</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c11">x</font>;<br/>
<b>set </b><font color="Maroon" title="c20">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#K2" title="FINSEQ_2:func.2">|-&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/>
<a NAME="E3:47_1_2_1_2_1"/><i><font color="Green" title="E54">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E4:47_1_2_1_2_1"/><i><font color="Green" title="E55">A28</font></i>: 
 <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="c13">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="c11">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>
 
<b>by </b><i><a class="txt" href="bspace.html#E3:47_1_2_1_2_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="ref" href="bspace.html#D11" target="_self" title="BSPACE:def.11">Def11</a></i>;<br/>
<a NAME="E5:47_1_2_1_2_1"/><i><font color="Green" title="E56">A29</font></i>: 
 <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#K2" title="FINSEQ_2:func.2">|-&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>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:47_1_2_1_2_1"/><i><font color="Green" title="E57">A30</font></i>: 
 <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="c13">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="c11">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#K2" title="FINSEQ_2:func.2">|-&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>
 
<b>by </b><i><a class="txt" href="bspace.html#E4:47_1_2_1_2_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="bspace.html#E5:47_1_2_1_2_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:47_1_2_1_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c13">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="c11">x</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c11">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#K2" title="FINSEQ_2:func.2">|-&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" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2_1_2_1_1"/><i><font color="Green" title="E58">A31</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">k</font>
 <b>and </b><br/><a NAME="E2:47_1_2_1_2_1_1"/><i><font color="Green" title="E59">A32</font></i>: 
<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="c13">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="c11">x</font></span>)</span>
 ;<br/>

<a NAME="E3:47_1_2_1_2_1_1"/><i><font color="Green" title="E60">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E4:47_1_2_1_2_1_1"/><i><font color="Green" title="E61">A34</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>
 
<b>by </b><i><a class="txt" href="bspace.html#E3:47_1_2_1_2_1_1"><i><font color="Green" title="E60">A33</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:47_1_2_1_2_1_1"/><i><font color="Green" title="E62">A35</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:47_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E63">A36</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN: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="c13">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bspace.html#E4:47_1_2_1_2_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="bspace.html#E2:47_1_2_1_2_1_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="bspace.html#E4:47_1_2_1_2_1_1"><i><font color="Green" title="E61">A34</font></i></a></i>;<br/>
<a NAME="E7:47_1_2_1_2_1_1"/><i><font color="Green" title="E64">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c13">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#K5" title="VECTSP_1:func.5">*</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>
 
<b>by </b><i><a class="txt" href="bspace.html#E6:47_1_2_1_2_1_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E8:47_1_2_1_2_1_1"/><i><font color="Green" title="E65">A38</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:47_1_2_1_2_1_1"/><i><font color="Green" title="E66">A39</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">F</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E4:47_1_2_1_2_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="bspace.html#E2:47_1_2_1_2_1_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="bspace.html#E5:47_1_2_1_2_1_1"><i><font color="Green" title="E62">A35</font></i></a>, <a class="txt" href="bspace.html#E8:47_1_2_1_2_1_1"><i><font color="Green" title="E65">A38</font></i></a></i>;<br/>
<a NAME="E10:47_1_2_1_2_1_1"/><i><font color="Green" title="E67">A40</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="bspace.html#E9:47_1_2_1_2_1_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E11:47_1_2_1_2_1_1"/><i><font color="Green" title="E68">A41</font></i>: 
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">F</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E9:47_1_2_1_2_1_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E12:47_1_2_1_2_1_1"/><i><font color="Green" title="E69">A42</font></i>: 
<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#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E2:47_1_2_1_2_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="bspace.html#E9:47_1_2_1_2_1_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="bspace.html#E10:47_1_2_1_2_1_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>reconsider </b><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="E14:47_1_2_1_2_1_1"/><i><font color="Green" title="E70">A43</font></i>: 
<font color="Maroon" title="c13">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#K2" title="GROUP_1:func.2">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E2:47_1_2_1_2_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="bspace.html#E10:47_1_2_1_2_1_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="bspace.html#E11:47_1_2_1_2_1_1"><i><font color="Green" title="E68">A41</font></i></a>, <a class="ref" href="bspace.html#T35" target="_self" title="BSPACE:th.35">Th35</a></i>;<br/>
<a NAME="E15:47_1_2_1_2_1_1"/><i><font color="Green" title="E71">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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>
 
<b>by </b><i><a class="txt" href="bspace.html#E7:47_1_2_1_2_1_1"><i><font color="Green" title="E64">A37</font></i></a>, <a class="txt" href="bspace.html#E14:47_1_2_1_2_1_1"><i><font color="Green" title="E70">A43</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E16:47_1_2_1_2_1_1"/><i><font color="Green" title="E72">A45</font></i>: 
 <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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_1_2_1_2_1_1_1"/>
 <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> <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>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:47_1_2_1_2_1_1_1"/>
<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> <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="c13">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1_2_1_2_1_1_1_1"/><i><font color="Green" title="E73">A46</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="subset_1.html#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>
 ;<br/>

<a NAME="E2:47_1_2_1_2_1_1_1_1"/><i><font color="Green" title="E74">A47</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1_1_1"><i><font color="Green" title="E73">A46</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">y</font> = <font color="Maroon" title="c23">y</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> <b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1_1_1"><i><font color="Green" title="E73">A46</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2_1_2_1_1_1_1_1"/><i><font color="Green" title="E75">A48</font></i>: 
not <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">m</font>
 ;<br/><a NAME="E2:47_1_2_1_2_1_1_1_1_1"/><i><font color="Green" title="E76">A49</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c24">y</font>
 <b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1_1_1"><i><font color="Green" title="E73">A46</font></i></a>, <a class="ref" href="ranknull.html#T25" title="RANKNULL:th.25">RANKNULL:25</a></i>;<br/><a NAME="E3:47_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c17">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c24">y</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> 
 <b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1_1_1_1"><i><font color="Green" title="E75">A48</font></i></a>, <a class="txt" href="bspace.html#E2:47_1_2_1_2_1_1_1_1_1"><i><font color="Green" title="E76">A49</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_1_2_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bspace.html#E2:47_1_2_1_2_1_1_1_1"><i><font color="Green" title="E74">A47</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">m</font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E17:47_1_2_1_2_1_1"/><i><font color="Green" title="E73">A50</font></i>: 
<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" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c22">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E2:47_1_2_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="bspace.html#E12:47_1_2_1_2_1_1"><i><font color="Green" title="E69">A42</font></i></a>, <a class="txt" href="bspace.html#E16:47_1_2_1_2_1_1"><i><font color="Green" title="E72">A45</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:47_1_2_1_2_1_1"/><i><font color="Green" title="E74">A51</font></i>: 
<font color="Maroon" title="c22">Fk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E12:47_1_2_1_2_1_1"><i><font color="Green" title="E69">A42</font></i></a>, <a class="txt" href="bspace.html#E16:47_1_2_1_2_1_1"><i><font color="Green" title="E72">A45</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green" title="E75">A52</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c11">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="c13">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="c11">x</font>
<b>by </b><i><a class="txt" href="bspace.html#E3:47_1_2_1_2_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="bspace.html#E4:47_1_2_1_2_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="bspace.html#E2:47_1_2_1_2_1_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="bspace.html#D11" target="_self" title="BSPACE:def.11">Def11</a></i>
<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> 
<b>by </b><i><a class="txt" href="bspace.html#E16:47_1_2_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="bspace.html#E15:47_1_2_1_2_1_1"><i><font color="Green" title="E71">A44</font></i></a>, <a class="txt" href="bspace.html#E17:47_1_2_1_2_1_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="bspace.html#E18:47_1_2_1_2_1_1"><i><font color="Green" title="E74">A51</font></i></a></i>
;<br/>
<a NAME="E20:47_1_2_1_2_1_1"/>
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN: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>
 
<b>by </b><i><a class="txt" href="bspace.html#E4:47_1_2_1_2_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="bspace.html#E1:47_1_2_1_2_1_1"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="bspace.html#E2:47_1_2_1_2_1_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="bspace.html#E5:47_1_2_1_2_1_1"><i><font color="Green" title="E62">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:47_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c11">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#K2" title="FINSEQ_2:func.2">|-&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>
 <b>by </b><i><a class="txt" href="bspace.html#E19:47_1_2_1_2_1_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:47_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c11">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#K2" title="FINSEQ_2:func.2">|-&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>
 <b>by </b><i><a class="txt" href="bspace.html#E6:47_1_2_1_2_1"><i><font color="Green" title="E57">A30</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c18">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><b> such that </b><br/><a NAME="E3:47_1_2_1_2"/><i><font color="Green" title="E53">A53</font></i>: 
<font color="Maroon" title="c18">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:47_1_2_1_2"/><i><font color="Green" title="E54">A54</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">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="c13">m</font>
 <b>and </b><br/><a NAME="E5:47_1_2_1_2"/><i><font color="Green" title="E55">A55</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" title="VECTSP_6:def.9">VECTSP_6:def 9</a></i>;<br/>
<a NAME="E6:47_1_2_1_2"/><i><font color="Green" title="E56">A56</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c18">F</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c11">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bspace.html#T34" target="_self" title="BSPACE:th.34">Th34</a></i>;<br/>
<a NAME="E7:47_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c11">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#K2" title="FINSEQ_2:func.2">|-&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>
 
<b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1_2"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="bspace.html#E3:47_1_2_1_2"><i><font color="Green" title="E53">A53</font></i></a>, <a class="txt" href="bspace.html#E4:47_1_2_1_2"><i><font color="Green" title="E54">A54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:47_1_2_1_2"/>
<font color="Maroon" title="c15">t</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="txt" href="bspace.html#E5:47_1_2_1_2"><i><font color="Green" title="E55">A55</font></i></a>, <a class="txt" href="bspace.html#E6:47_1_2_1_2"><i><font color="Green" title="E56">A56</font></i></a>, <a class="ref" href="bspace.html#T5" target="_self" title="BSPACE:th.5">Th5</a>, <a class="ref" href="matrix_3.html#T13" title="MATRIX_3:th.13">MATRIX_3:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:47_1_2_1"/><i><font color="Green" title="E52">A57</font></i>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_2_1_3"/><i><font color="Green" title="E53">A58</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c12">g</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c12">g</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T43" title="VECTSP_6:th.43">VECTSP_6:43</a></i>;<br/>
<a NAME="E2:47_1_2_1_3"/><i><font color="Green" title="E54">A59</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:47_1_2_1_3"/><i><font color="Green" title="E55">A60</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c12">g</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E2:47_1_2_1_3"><i><font color="Green" title="E54">A59</font></i></a>, <a class="ref" href="ranknull.html#T25" title="RANKNULL:th.25">RANKNULL:25</a></i>;<br/>
<a NAME="E4:47_1_2_1_3"/><i><font color="Green" title="E56">A61</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c12">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> 
 
<b>by </b><i><a class="txt" href="bspace.html#E5:47_1_2_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/>
<a NAME="E5:47_1_2_1_3"/><i><font color="Green" title="E57">A62</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="bspace.html#E4:47_1_2_1_3"><i><font color="Green" title="E56">A61</font></i></a>, <a class="ref" href="bspace.html#T5" target="_self" title="BSPACE:th.5">Th5</a>, <a class="ref" href="bspace.html#T6" target="_self" title="BSPACE:th.6">Th6</a>, <a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">o</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  ;<br/>
<b>thus </b><a NAME="E7:47_1_2_1_3"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="bspace.html#E1:47_1_2_1_3"><i><font color="Green" title="E53">A58</font></i></a>, <a class="txt" href="bspace.html#E3:47_1_2_1_3"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="bspace.html#E5:47_1_2_1_3"><i><font color="Green" title="E57">A62</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E30:47_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bspace.html#E13:47_1_2_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="bspace.html#E15:47_1_2_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="bspace.html#E27:47_1_2_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="bspace.html#E28:47_1_2_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="bspace.html#E29:47_1_2_1"><i><font color="Green" title="E52">A57</font></i></a>, <a class="ref" href="bspace.html#T5" target="_self" title="BSPACE:th.5">Th5</a>, <a class="ref" href="bspace.html#T6" target="_self" title="BSPACE:th.6">Th6</a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:47_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>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D1" title="VECTSP_7:def.1">VECTSP_7:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
