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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c4">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>;<br/>





<a NAME="E1:10"/><i><font color="Green" title="E4">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E2:10"/><i><font color="Green" title="E5">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3" title="MATRIX_3:def.3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green" title="E6">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E4:10"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E7">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 ;<br/>

<a NAME="E2:10_1"/><i><font color="Green" title="E8">A5</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a>, <a class="txt" href="matrix_5.html#E1:10"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<a NAME="E3:10_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a>, <a class="txt" href="matrix_5.html#E3:10"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D3" title="MATRIX_3:def.3">MATRIX_3:def 3</a>, <a class="txt" href="matrix_5.html#E1:10_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="matrix_5.html#E2:10_1"><i><font color="Green" title="E8">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a>, <a class="txt" href="matrix_5.html#E1:10_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="matrix_5.html#E2:10_1"><i><font color="Green" title="E8">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a>, <a class="txt" href="matrix_5.html#E1:10_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="matrix_5.html#E2:10_1"><i><font color="Green" title="E8">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E5:10_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a>, <a class="txt" href="matrix_5.html#E1:10_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="matrix_5.html#E2:10_1"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a>, <a class="txt" href="matrix_5.html#E1:10"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="matrix_5.html#E2:10"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="matrix_5.html#E3:10"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>


</div>
