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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">K</font> be    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) or ( not <font color="Maroon">a<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> ) );<br/>
<a NAME="E1:10"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> or  not <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:10_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">K</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:10_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_2"/>
not <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:10"/>
 ex <font color="Olive">L</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">L</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font><b> such that </b><br/><a NAME="E4:10"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 ;<br/>
<a NAME="E5:10"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:10_2"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:10_2"/><b>then </b>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>,<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>] &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> ) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>] )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:10_2"/><b>then </b>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>,<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>] &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> ) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>] )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/><b>hence </b><a NAME="E4:10_2"/>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:10"/>
<font color="Maroon">L</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="vectsp_6.html#D4">VECTSP_6:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:10_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/><a NAME="E4:10_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>,<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>] )
 <b>by </b><i>, , , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:10_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E11:10"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> ;<br/>
<b>reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/><a NAME="E2:10_4"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>] )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:10_4"/>
<font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:10"/><b>then </b>
<font color="Maroon">K'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E16:10"/>
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> )
 <b>by </b><i>, </i>;<br/>


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