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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:65"/><i><font color="Green">E28</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:65"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:65"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:65"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:65"/><i><font color="Green">E32</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E7:65"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T6">EUCLID_2:6</a>, <a class="txt" href="matrixr1.html#E4:65"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrixr1.html#E5:65"><i><font color="Green">E31</font></i></a>, <a class="txt" href="matrixr1.html#E1:65"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E4:65"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:65"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <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"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:65_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T25" target="_self">Th25</a>, <a class="txt" href="matrixr1.html#E1:65_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E3:65_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E3:65"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:65_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E2:65_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E5:65_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E2:65_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixr1.html#E1:65"><i><font color="Green">E28</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:65_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T25" target="_self">Th25</a>, <a class="txt" href="matrixr1.html#E3:65_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrixr1.html#E4:65_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:65_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a>, <a class="txt" href="matrixr1.html#E1:65_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrixr1.html#E8:65"><i><font color="Green">E34</font></i></a>, <a class="txt" href="matrixr1.html#E6:65"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:65"/>
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixr1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixr1.html#E4:65"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrixr1.html#E6:65"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrixr1.html#E7:65"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
