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<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>set </b><font color="Maroon" title="c2">V</font> =  <a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c3">S</font> =  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:48"/>
for <font color="Olive" title="b1">v</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="c1">X</font></span>)</span> holds  <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">v</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="bspace.html#E3:48_1"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E7:48_1"/><i><font color="Green" title="E38">A3</font></i>: 
 <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">A</font>
 <b>by </b><i><a class="ref" href="bspace.html#T39" target="_self" title="BSPACE:th.39">Th39</a></i>;<br/>
<b>thus </b><a NAME="E8:48_1"/>
<font color="Maroon" title="c4">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="bspace.html#E4:48_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="bspace.html#E7:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T40" title="VECTSP_4:th.40">VECTSP_4:40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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