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

<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">ea</font> = <font color="Maroon" title="c1">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  ;<br/>
<i><font color="Green" title="E20">A2</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="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T27" target="_self" title="MATRIXR1:th.27">Th27</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="matrixr1.html#D9" target="_self" title="MATRIXR1:def.9">Def9</a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a></i>
;<br/>
<i><font color="Green" title="E21">A3</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T27" target="_self" title="MATRIXR1:th.27">Th27</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrixr1.html#D9" target="_self" title="MATRIXR1:def.9">Def9</a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a></i>
;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E22">A4</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="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</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="c2">x</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8" title="EUCLID_2:th.8">EUCLID_2:8</a></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:53"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:53_3"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
 ;<br/>

<a NAME="E2:53_3"/><b>then </b><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E5:53"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:53_3"/><i><font color="Green" title="E26">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</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="c2">x</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b1">j2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">x</font> holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> ) )
 
<b>by </b><i><a class="ref" href="matrixr1.html#D9" target="_self" title="MATRIXR1:def.9">Def9</a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E4:53_3"/><b>then </b><i><font color="Green" title="E27">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:53_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <span class="p3">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E3:53_3"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:53_3"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">j</font>,1</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"><a href="matrixr1.html#K1" title="MATRIXR1:func.1">MXR2MXF</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E2:53_3"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="matrixr1.html#E4:53_3"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:53_3"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E1:53_3"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="matrixr1.html#E3:53"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="matrixr1.html#E5:53"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:53_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E11:53_3"/><i><font color="Green" title="E30">A12</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s</font> <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="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a>, <a class="txt" href="matrixr1.html#E3:53"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:53_3"/><i><font color="Green" title="E31">A13</font></i>: 
for <font color="Olive" title="b1">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">s2</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1" title="MATRIX_1:def.1">MATRIX_1:def 1</a></i>;<br/>
<a NAME="E14:53_3"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E7:53_3"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E16:53_3"/><i><font color="Green" title="E32">A14</font></i>: 
( <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="matrixr1.html#E13:53_3"><i><font color="Green" title="E31">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">s3</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E18:53_3"/><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c9">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="matrixr1.html#E13:53_3"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="matrixr1.html#E11:53_3"><i><font color="Green" title="E30">A12</font></i></a></i>;<br/>
<i><font color="Green" title="E34">A16</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T27" target="_self" title="MATRIXR1:th.27">Th27</a>, <a class="txt" href="matrixr1.html#E16:53_3"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="matrixr1.html#E18:53_3"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="matrixr1.html#E11:53_3"><i><font color="Green" title="E30">A12</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrixr1.html#D9" target="_self" title="MATRIXR1:def.9">Def9</a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E20:53_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">j</font>,1</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"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E6:53_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="matrixr1.html#T28" target="_self" title="MATRIXR1:th.28">Th28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E22:53_3"/><i><font color="Green" title="E35">A17</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">p3</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E24:53_3"/><i><font color="Green" title="E36">A18</font></i>: 
( <font color="Maroon" title="c11">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p3</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrixr1.html#E6:53_3"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/>
<a NAME="E25:53_3"/><i><font color="Green" title="E37">A19</font></i>: 
<span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D9" target="_self" title="MATRIXR1:def.9">Def9</a>, <a class="txt" href="matrixr1.html#E1:53_3"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="matrixr1.html#E6:53"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c4">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green" title="E38">A20</font></i>: <font color="Maroon" title="c5">q</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 = 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">j</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T29" target="_self" title="MATRIXR1:th.29">Th29</a>, <a class="txt" href="matrixr1.html#E6:53_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="matrixr1.html#E22:53_3"><i><font color="Green" title="E35">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a>, <a class="txt" href="matrixr1.html#E24:53_3"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="matrixr1.html#E25:53_3"><i><font color="Green" title="E37">A19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E28:53_3"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> holds <br/><font color="Maroon" title="c5">q</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_3_3"/><i><font color="Green" title="E39">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/>

<a NAME="E2:53_3_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E3:53_3_3"/><b>then </b>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="matrixr1.html#E1:53_3_3"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E4:53_3_3"/>
<font color="Maroon" title="c5">q</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="matrixr1.html#E27:53_3"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E29:53_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="txt" href="matrixr1.html#E19:53_3"><i><font color="Green" title="E34">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
 <a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">a</font> <a href="matrixr1.html#K7" title="MATRIXR1:func.7">*</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K9" title="MATRIXR1:func.9">ColVec2Mx</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixr1.html#D9" target="_self" title="MATRIXR1:def.9">Def9</a>, <a class="txt" href="matrixr1.html#E3:53"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="matrixr1.html#E4:53"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="matrixr1.html#E1:53"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="matrixr1.html#E5:53"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>


</div>
