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<div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>reconsider </b><font color="Maroon">ea</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#K3">F_Real</a>  ;<br/>
<i><font color="Green">E31</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i/>
;<br/>
<i><font color="Green">E54</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<a NAME="E4:60"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:60"/>
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:60_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>

<a NAME="E2:60_3"/><i><font color="Green">E58</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">j2</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Olive">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j2</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:60_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:60_3"/><b>then </b><i><font color="Green">E59</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:60_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p3">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:60_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:60_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:60_3"/><i><font color="Green">E63</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:60_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , <a class="ref" href="matrixr1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E11:60_3"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, , <a class="txt" href="matrixr1.html#E2:60"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:60_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</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/>
<font color="Maroon">q</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrixr1.html#E2:60"><i><font color="Green">E31</font></i></a>, <a class="ref" href="matrixr1.html#T2" target="_self">Th2</a>, <a class="ref" href="matrixr1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<b>hence </b><a NAME="E16:60_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="matrixr1.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:60"/>
 <a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="matrixr1.html#T1" target="_self">Th1</a></i>;<br/>


</div>
