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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">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="c1">X</font> st  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c1">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> <font color="Maroon" title="c2">A</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">V</font> =  <a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $1 is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> implies  ex <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="c1">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> $1 );<br/>
<a NAME="E1:48"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">l</font> =  <a href="vectsp_6.html#K2" title="VECTSP_6:func.2">ZeroLC</a> <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> as    <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="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T26" title="VECTSP_6:th.26">VECTSP_6:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:48_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c1">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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">l</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E3:48_1"/>
 <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="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T41" title="VECTSP_6:th.41">VECTSP_6:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:48_1"/>
 <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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:48"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B</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> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E2:48_2"/>
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E3:48_2"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<span class="kw">assume </span><a NAME="E4:48_2"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c1">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> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">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="c1">X</font><span class="kw"> such that </span><br/><a NAME="E7:48_2"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E37">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_2_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_2_1_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c1">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> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">l</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">thus </span><a NAME="E2:48_2_1_1"/>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E1:48_2_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_2_1_2"/>
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c1">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> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:48_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">z</font> =  <a href="vectsp_6.html#K2" title="VECTSP_6:func.2">ZeroLC</a> <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> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T26" title="VECTSP_6:th.26">VECTSP_6:26</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_2"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">f</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">m</font> = <font color="Maroon" title="c9">z</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> </span>)</span>;<br/><a NAME="E6:48_2_1_2"/>
<font color="Maroon" title="c9">z</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> </span>)</span> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p2">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T23" title="RANKNULL:th.23">RANKNULL:23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c9">z</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c10">f</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> </span>)</span> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/><a NAME="E8:48_2_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></span>;<br/><a NAME="E9:48_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>;<br/><a NAME="E10:48_2_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E11:48_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">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="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E12:48_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">m</font> is    <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="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T25" title="VECTSP_6:th.25">VECTSP_6:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c8">l</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c13">m</font> as    <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="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T52" title="VECTSP_6:th.52">VECTSP_6:52</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c14">n</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</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="c13">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T77" title="VECTSP_6:th.77">VECTSP_6:77</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="bspace.html#K10" title="BSPACE:func.10">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T43" title="VECTSP_6:th.43">VECTSP_6:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_2_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="vectsp_1.html#D29" title="VECTSP_1:def.29">VECTSP_1:def 29</a></span>
<br/>.= 
<font color="Maroon" title="c7">B</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c11">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="bspace.html#D5" target="_self" title="BSPACE:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T101" title="XBOOLE_1:th.101">XBOOLE_1:101</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2_1_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

;<br/><span class="kw">hence </span><a NAME="E15:48_2_1_2"/>
 <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> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:48"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E4:48"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E36">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:48"/>
 ex <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="c1">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> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
