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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</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">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</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">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E6:18"/>
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"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</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"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E42</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"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:18_1"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</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"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T55">MATRIX_4:55</a>, </i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green">E44</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T55">MATRIX_4:55</a>, </i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green">E45</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T55">MATRIX_4:55</a>, </i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E56</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T55">MATRIX_4:55</a>, </i>;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</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="matrix_3.html#D3">MATRIX_3:def 3</a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</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="matrix_3.html#D5">MATRIX_3:def 5</a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a>, , , , </i>
<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</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="matrix_3.html#D3">MATRIX_3:def 3</a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a>, , </i>
;<br/>
<b>hence </b><a NAME="E8:18_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</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"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:18"/>
<font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, , , , </i>;<br/>


</div>
