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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">V1</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">F</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V1</font>;<br/><b>let </b><font color="Maroon">KL</font> be    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V1</font>;<br/>







<b>assume </b><a NAME="E1:34"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E44</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> =  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>, <font color="Maroon">G2</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V1</font> <b>by </b><i><a class="ref" href="finseq_3.html#T93">FINSEQ_3:93</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T94">FINSEQ_3:94</a></i>;<br/>
<i><font color="Green">E52</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G1</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>
<br/>.= 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E7:34"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span>
 ;<br/>

<a NAME="E2:34_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E3:34_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:34_2"/><b>then </b>
<font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:34_2"/><b>then </b>
<font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/>
<a NAME="E6:34_2"/><b>then </b>
not <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:34_2"/><b>then </b><i><font color="Green">E57</font></i>: 
not <font color="Maroon">G2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V1</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p1</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E10:34"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T124">FINSEQ_3:124</a></i>;<br/>
<a NAME="E11:34"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E12:34"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<a NAME="E14:34"/>
<font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">KL</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V1</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="vectsp_6.html#D9">VECTSP_6:def 9</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><br/>


</div>
