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

<b>let </b><font color="Maroon" title="c1">n</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>let </b><font color="Maroon" title="c2">L</font>, <font color="Maroon" title="c3">L1</font>, <font color="Maroon" title="c4">L2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c5">P</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>;<br/>









<b>assume </b><a NAME="E1:125"/><i><font color="Green" title="E103">A1</font></i>: 
( <font color="Maroon" title="c2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> )
 ;<br/>

<a NAME="E2:125"/><b>then </b><i><font color="Green" title="E104">A2</font></i>: 
( <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  &amp; <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c4">L2</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  &amp; <font color="Maroon" title="c3">L1</font>,<font color="Maroon" title="c4">L2</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T101" target="_self" title="EUCLIDLP:th.101">Th101</a></i>;<br/>
<b>assume </b><a NAME="E3:125"/><i><font color="Green" title="E105">A3</font></i>: 
( <font color="Maroon" title="c2">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L1</font> &amp; <font color="Maroon" title="c2">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font> )
 ;<br/>

<a NAME="E4:125"/><b>then </b><i><font color="Green" title="E106">A4</font></i>: 
( <font color="Maroon" title="c2">L</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L1</font> &amp; <font color="Maroon" title="c2">L</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">L2</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E2:125"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="euclidlp.html#T114" target="_self" title="EUCLIDLP:th.114">Th114</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:125"/><i><font color="Green" title="E107">A5</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">x2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E8:125"/><i><font color="Green" title="E108">A6</font></i>: 
( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:125"><i><font color="Green" title="E106">A4</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<a NAME="E9:125"/><i><font color="Green" title="E109">A7</font></i>: 
( <font color="Maroon" title="c2">L</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c3">L1</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c4">L2</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:125"><i><font color="Green" title="E105">A3</font></i></a>, <a class="ref" href="euclidlp.html#T72" target="_self" title="EUCLIDLP:th.72">Th72</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> or <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1_1"><b>case </b></a><a NAME="E1:125_1_1_1"/><i><font color="Green" title="E110">A8</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 ;<br/><div class="add"><a NAME="E2:125_1_1_1"/>
<font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:125"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:125"><i><font color="Green" title="E105">A3</font></i></a>, <a class="txt" href="euclidlp.html#E6:125"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:125"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="euclidlp.html#E1:125_1_1_1"><i><font color="Green" title="E110">A8</font></i></a>, <a class="ref" href="euclidlp.html#T113" target="_self" title="EUCLIDLP:th.113">Th113</a></i>;<br/><b>hence </b><a NAME="E3:125_1_1_1"/>
<font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E9:125"><i><font color="Green" title="E109">A7</font></i></a>, <a class="ref" href="euclidlp.html#T70" target="_self" title="EUCLIDLP:th.70">Th70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1_2"><b>case </b></a><a NAME="E1:125_1_1_2"/><i><font color="Green" title="E110">A9</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">x0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:125_1_1_2"/><i><font color="Green" title="E111">A10</font></i>: 
( <font color="Maroon" title="c8">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c8">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E6:125"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="euclidlp.html#E9:125"><i><font color="Green" title="E109">A7</font></i></a>, <a class="ref" href="euclidlp.html#T58" target="_self" title="EUCLIDLP:th.58">Th58</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:125_1_1_2"/><i><font color="Green" title="E112">A11</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:125"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="euclidlp.html#E9:125"><i><font color="Green" title="E109">A7</font></i></a>, <a class="ref" href="euclidlp.html#T58" target="_self" title="EUCLIDLP:th.58">Th58</a></i>;<br/><a NAME="E6:125_1_1_2"/><i><font color="Green" title="E113">A12</font></i>: 
<font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c7">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:125"><i><font color="Green" title="E105">A3</font></i></a>, <a class="txt" href="euclidlp.html#E6:125"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:125"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green" title="E110">A9</font></i></a>, <a class="txt" href="euclidlp.html#E3:125_1_1_2"><i><font color="Green" title="E111">A10</font></i></a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th79</a></i>;<br/><a NAME="E7:125_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a> 
 <b>by </b><i><a class="ref" href="euclidlp.html#T50" target="_self" title="EUCLIDLP:th.50">Th50</a></i>;<br/><a NAME="E8:125_1_1_2"/><b>then </b><i><font color="Green" title="E114">A13</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:125"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="euclidlp.html#E6:125"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:125"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="euclidlp.html#E3:125_1_1_2"><i><font color="Green" title="E111">A10</font></i></a>, <a class="ref" href="euclidlp.html#T97" target="_self" title="EUCLIDLP:th.97">Th97</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c11">a3</font>, <font color="Maroon" title="c12">a2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:125_1_1_2"/><i><font color="Green" title="E115">A14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c11">a3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c12">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c9">x</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="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">x0</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E5:125_1_1_2"><i><font color="Green" title="E112">A11</font></i></a>, <a class="txt" href="euclidlp.html#E1:125"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="euclidlp.html#E8:125_1_1_2"><i><font color="Green" title="E114">A13</font></i></a>, <a class="ref" href="euclidlp.html#T93" target="_self" title="EUCLIDLP:th.93">Th93</a></i>;<br/><a NAME="E11:125_1_1_2"/><i><font color="Green" title="E116">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a2</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c11">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:125_1_1_2"><i><font color="Green" title="E115">A14</font></i></a></i>;<br/><i><font color="Green" title="E117">A16</font></i>: <font color="Maroon" title="c9">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">x0</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font>
<b>by </b><i><a class="txt" href="euclidlp.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a>, <a class="txt" href="euclidlp.html#E10:125_1_1_2"><i><font color="Green" title="E115">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E11:125_1_1_2"><i><font color="Green" title="E116">A15</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
;<br/><a NAME="E13:125_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:125"><i><font color="Green" title="E105">A3</font></i></a>, <a class="txt" href="euclidlp.html#E6:125"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:125"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green" title="E110">A9</font></i></a>, <a class="txt" href="euclidlp.html#E5:125_1_1_2"><i><font color="Green" title="E112">A11</font></i></a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th79</a></i>;<br/><a NAME="E14:125_1_1_2"/><b>then </b><i><font color="Green" title="E118">A17</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/><a NAME="E15:125_1_1_2"/>
<font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a> 
 <b>by </b><i><a class="txt" href="euclidlp.html#E6:125_1_1_2"><i><font color="Green" title="E113">A12</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/><a NAME="E16:125_1_1_2"/><b>then </b><i><font color="Green" title="E119">A18</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="euclid_4.html#D6" title="EUCLID_4:def.6">EUCLID_4:def 6</a></i>;<br/><i><font color="Green" title="E120">A19</font></i>: <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E13"><i><font color="Green" title="E8">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="seq_1.html#T36" title="SEQ_1:th.36">SEQ_1:36</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E13"><i><font color="Green" title="E8">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T6" title="EUCLID_4:th.6">EUCLID_4:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T27" title="EUCLID_4:th.27">EUCLID_4:27</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="euclidlp.html#E16:125_1_1_2"><i><font color="Green" title="E119">A18</font></i></a></i>
;<br/><i><font color="Green" title="E121">A20</font></i>: <font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> = 
<font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
<br/>.= 
<font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font>
<b>by </b><i><a class="txt" href="euclidlp.html#E5"><i><font color="Green" title="E5">Lm4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font>
<b>by </b><i><a class="ref" href="euclidlp.html#T9" target="_self" title="EUCLIDLP:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T10" target="_self" title="EUCLIDLP:th.10">Th10</a></i>
;<br/><i><font color="Green" title="E122">A21</font></i>: 0 = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E14:125_1_1_2"><i><font color="Green" title="E118">A17</font></i></a>, <a class="ref" href="euclid_4.html#D6" title="EUCLID_4:def.6">EUCLID_4:def 6</a>, <a class="txt" href="euclidlp.html#E12:125_1_1_2"><i><font color="Green" title="E117">A16</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T25" title="EUCLID_4:th.25">EUCLID_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T25" title="EUCLID_4:th.25">EUCLID_4:25</a>, <a class="txt" href="euclidlp.html#E18:125_1_1_2"><i><font color="Green" title="E121">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c11">a3</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E17:125_1_1_2"><i><font color="Green" title="E120">A19</font></i></a></i>
;<br/><a NAME="E20:125_1_1_2"/>
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green" title="E110">A9</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/><a NAME="E21:125_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="euclid_4.html#T22" title="EUCLID_4:th.22">EUCLID_4:22</a></i>;<br/><a NAME="E22:125_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c11">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="euclidlp.html#E19:125_1_1_2"><i><font color="Green" title="E122">A21</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E23:125_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c10">a1</font>
 ;<br/><b>then </b><i><font color="Green" title="E123">A22</font></i>: <font color="Maroon" title="c9">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> = 
<font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a>, <a class="txt" href="euclidlp.html#E12:125_1_1_2"><i><font color="Green" title="E117">A16</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x0</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T23" target="_self" title="EUCLIDLP:th.23">Th23</a></i>
<br/>.= 
<font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></i>
<br/>.= 
<font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
;<br/><a NAME="E25:125_1_1_2"/>
( <font color="Maroon" title="c9">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:125_1_1_2"><i><font color="Green" title="E111">A10</font></i></a>, <a class="txt" href="euclidlp.html#E5:125_1_1_2"><i><font color="Green" title="E112">A11</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/><a NAME="E26:125_1_1_2"/><b>then </b><i><font color="Green" title="E124">A23</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x0</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E24:125_1_1_2"><i><font color="Green" title="E123">A22</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/><a NAME="E27:125_1_1_2"/>
( <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c4">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E6:125"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:125"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="euclidlp.html#E3:125_1_1_2"><i><font color="Green" title="E111">A10</font></i></a>, <a class="txt" href="euclidlp.html#E5:125_1_1_2"><i><font color="Green" title="E112">A11</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></i>;<br/><b>hence </b><a NAME="E28:125_1_1_2"/>
<font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E26:125_1_1_2"><i><font color="Green" title="E124">A23</font></i></a>, <a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:125"/>
<font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font>
 ;<br/>


</div>
