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

<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  not $1 is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> or ( $1 <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="c4">F</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) or ( not $1 <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="c4">F</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> ) );<br/>
<a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c5">x</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1"/>
( <font color="Maroon" title="c5">x</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="c4">F</font> or  not <font color="Maroon" title="c5">x</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="c4">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1"/>
<font color="Maroon" title="c5">x</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="c4">F</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x'</font>]
 ;<br/><b>hence </b><a NAME="E3:6_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_2"/>
not <font color="Maroon" title="c5">x</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="c4">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:6"/>
 ex <font color="Olive" title="b1">K</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="vectsp_9.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">K</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font><b> such that </b><br/><a NAME="E4:6"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">R</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<div><i><font color="Green" title="E8">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><b>assume </b><a NAME="E1:6_2"/><i><font color="Green" title="E9">A4</font></i>: 
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_1"/>
(  not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> or  not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font> )
 </span><b>by </b><i><a class="txt" href="vectsp_9.html#E1:6_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><b>suppose </b></a><a NAME="E1:6_2_1_1"/>
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_2_1_1"/>
<font color="Maroon" title="c5">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:6"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_2"><b>suppose </b></a><a NAME="E1:6_2_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font>
 ;<br/><div class="add"><a NAME="E2:6_2_1_2"/>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>] ) or ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">v</font>, <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>] ) )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:6"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:6_2_1_2"/>
<font color="Maroon" title="c5">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E1:6_2_1_2"><i><font color="Green" title="E10">A5</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></div><b>end;</b></div></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c5">K</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">K</font> = <font color="Maroon" title="c7">K</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E6:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="vectsp_6.html#D4" title="VECTSP_6:def.4">VECTSP_6:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">K</font>
;<br/>

<a NAME="E9:6"/><i><font color="Green" title="E9">A6</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:6_3_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">K</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">v'</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:6_3_1"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c10">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> &amp; <font color="Maroon" title="c8">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19" title="VECTSP_6:th.19">VECTSP_6:19</a></i>;<br/><b>thus </b><a NAME="E4:6_3_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E6:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="vectsp_9.html#E3:6_3_1"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:6_3"/><b>then </b><i><font color="Green" title="E10">A8</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:6_3_2"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>
 ;<br/><a NAME="E2:6_3_2"/><b>then </b><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">v'</font> = <font color="Maroon" title="c9">v</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> ;<br/><a NAME="E4:6_3_2"/>
( <font color="Maroon" title="c8">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v'</font> &amp; <font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="vectsp_9.html#E2:6_3_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/><b>hence </b><a NAME="E5:6_3_2"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">K</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19" title="VECTSP_6:th.19">VECTSP_6:19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:6_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</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="c8">K</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:6_3"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:6_3"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:6"/>
<font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>set </b><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font>;<br/>
<a NAME="E1:6_4"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">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="c4">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">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="c4">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="E2:6_4"/><b>then </b><i><font color="Green" title="E10">A10</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="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_4_1"/><i><font color="Green" title="E11">A11</font></i>: 
<font color="Maroon" title="c11">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="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font>;<br/><a NAME="E2:6_4_1"/>
<font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:6_4"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="vectsp_9.html#E1:6_4_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:6_4_1"/><b>then </b><i><font color="Green" title="E12">A12</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c8">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span>] )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="vectsp_9.html#E2:6_4"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="vectsp_9.html#E1:6_4_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:6_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:6_4"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="vectsp_9.html#E1:6_4_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/><b>hence </b><a NAME="E5:6_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:6_4_1"><i><font color="Green" title="E12">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_4"/>
<font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:6_4"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:6"/>
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E9:6"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>


</div>
