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

<b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> <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">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrixr1.html#K5" title="MATRIXR1:func.5">-</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> <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">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrixr1.html#K5" title="MATRIXR1:func.5">-</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span></span><br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green" title="E23">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">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</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="c1">x1</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T7" title="EUCLID_2:th.7">EUCLID_2:7</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font> <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">x2</font>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green" title="E25">A4</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</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="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green" title="E26">A5</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</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="c1">x1</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E27">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</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">x2</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<a NAME="E7:25"/><i><font color="Green" title="E28">A7</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">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</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="c1">x1</font>
 
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:25"/><i><font color="Green" title="E29">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#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<a NAME="E9:25"/><i><font color="Green" title="E30">A9</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#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<a NAME="E10:25"/><i><font color="Green" title="E31">A10</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#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<a NAME="E11:25"/><i><font color="Green" title="E32">A11</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></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> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="#E5:25"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="#E8:25"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="#E9:25"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="euclid_2.html#T7" title="EUCLID_2:th.7">EUCLID_2:7</a>, <a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a></i>;<br/>
<a NAME="E12:25"/><i><font color="Green" title="E33">A12</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</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> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E5:25"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="#E6:25"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="#E9:25"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E10:25"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="matrix_4.html#T55" title="MATRIX_4:th.55">MATRIX_4:55</a></i>;<br/>
<i><font color="Green" title="E34">A13</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#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>
;<br/>
<i><font color="Green" title="E35">A14</font></i>:  <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#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E15:25"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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 <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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</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></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:25_3"/><i><font color="Green" title="E36">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span><br/>

<a NAME="E2:25_3"/><b>then </b><i><font color="Green" title="E37">A16</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; <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="c1">x1</font> )
 
<b>by </b><i><a class="txt" href="#E13:25"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="#E14:25"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:25_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:25_3"/><b>then </b><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E6:25_3"/><i><font color="Green" title="E39">A18</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="#E11:25"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="#E1:25_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E7:25_3"/><i><font color="Green" title="E40">A19</font></i>: 
<font color="Maroon" title="c5">p</font> <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">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="#E7:25"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="#E2:25_3"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="#E4:25_3"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="#E6:25_3"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">q1</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="E9:25_3"/><i><font color="Green" title="E41">A20</font></i>: 
( <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="#E1:25_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E10:25_3"/><i><font color="Green" title="E42">A21</font></i>: 
<font color="Maroon" title="c6">q1</font> <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> <font color="Maroon" title="c1">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="#E2:25_3"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="#E4:25_3"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="#E9:25_3"><i><font color="Green" title="E41">A20</font></i></a>, <a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q2</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="E12:25_3"/><i><font color="Green" title="E43">A22</font></i>: 
( <font color="Maroon" title="c7">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="#E12:25"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="#E1:25_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E13:25_3"/>
<font color="Maroon" title="c7">q2</font> <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> <font color="Maroon" title="c2">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="#E3:25"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E2:25_3"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="#E4:25_3"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="#E12:25_3"><i><font color="Green" title="E43">A22</font></i></a>, <a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E14:25_3"/>
<span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E6:25_3"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="#E7:25_3"><i><font color="Green" title="E40">A19</font></i></a>, <a class="txt" href="#E9:25_3"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="#E10:25_3"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="#E12:25_3"><i><font color="Green" title="E43">A22</font></i></a>, <a class="txt" href="matrixr2.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:25"/>
 <a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="matrixr1.html#K5" title="MATRIXR1:func.5">-</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">x2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="#E5:25"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="#E6:25"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="#E8:25"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="#E9:25"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E10:25"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="matrixr2.html#T22" target="_self" title="MATRIXR2:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
