<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T6">EUCLID_2:6</a></i>;<br/>
<a NAME="E3:59"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:59"/><i><font color="Green">E56</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> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:59"/><i><font color="Green">E57</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">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:59"/><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">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:59"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:59"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E9:59"/><i><font color="Green">E64</font></i>: 
 <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">x1</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E10:59"/><i><font color="Green">E69</font></i>: 
 <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">x2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E11:59"/><i><font color="Green">E70</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</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">x1</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="matrixr1.html#E1:59"><i><font color="Green">E31</font></i></a>, <a class="ref" href="matrixr1.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_4.html#T55">MATRIX_4:55</a>, </i>;<br/>
<a NAME="E12:59"/><i><font color="Green">E94</font></i>: 
 <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">x2</font></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">x1</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="matrixr1.html#T2" target="_self">Th2</a>, <a class="ref" href="matrixr1.html#T3" target="_self">Th3</a>, <a class="ref" href="matrix_4.html#T55">MATRIX_4:55</a></i>;<br/>
<i><font color="Green">E96</font></i>:  <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">x1</font></span>)</span> = 
 <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">x1</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 href="finseq_1.html#K2">Seg</a> 1
<b>by </b><i/>
;<br/>
<i><font color="Green">E98</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x1</font></span>)</span></span>)</span> = 
 <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">x1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E15:59"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</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">x1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="binop_2.html#K9">+</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">x2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:59_3"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<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"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x1</font></span>)</span>
 ;<br/>

<a NAME="E2:59_3"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:59_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:59_3"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>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="E6:59_3"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<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/>
<a NAME="E7:59_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>consider </b><font color="Maroon">q1</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="E9:59_3"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</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/>
<a NAME="E10:59_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T14" target="_self">Th14</a>, , , </i>;<br/>
<b>consider </b><font color="Maroon">q2</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="E12:59_3"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</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/>
<a NAME="E13:59_3"/>
<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , , <a class="ref" href="matrixr1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E14:59_3"/>
<span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</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">x1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K9">+</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">x2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="matrixr1.html#T14" target="_self">Th14</a>, , , <a class="ref" href="matrixr1.html#T16" target="_self">Th16</a>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:59"/>
 <a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x1</font></span>)</span> <a href="matrixr1.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" target="_self">LineVec2Mx</a> <font color="Maroon">x2</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="matrixr1.html#E1:59"><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/>


</div>
