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<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V2</font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">b2</font> be    <a href="matrlin.html#M1" target="_self">OrdBasis</a> of <font color="Maroon">V2</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V2</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V2</font>;<br/>









<b>assume </b><a NAME="E1:53"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">v1</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font>
 ;<br/>

<b>consider </b><font color="Maroon">KL1</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V2</font><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL1</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font> &amp; ( for <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">KL2</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V2</font><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL2</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font> &amp; ( for <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E6:53"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E1:53"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrlin.html#E3:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> )
 ;<br/><b>hence </b><font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">t</font>
<b>by </b><i>, <a class="txt" href="matrlin.html#E1:53"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrlin.html#E3:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="matrlin.html#T4" target="_self">Th4</a></i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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">V2</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1"/>
(  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font></span>)</span> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:53_2_1_1"/><b>then </b>
<font color="Maroon">KL2</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">K</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/><b>hence </b><a NAME="E4:53_2_1_1"/>
<font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_2_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E5:53_2_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:53_2_1_2"/><i><font color="Green">E66</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:53_2_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:53_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:53_2_1_2"/>
<font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:53"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:53"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrlin.html#E3:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="vectsp_6.html#D10">VECTSP_6:def 10</a></i>;<br/>


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