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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font>, <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</font> being   <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/> for <font color="Olive" title="b4">P</font> being   <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>  st <font color="Olive" title="b1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b1">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b2">L1</font> &amp; <font color="Olive" title="b1">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b3">L2</font> holds <br/><font color="Olive" title="b2">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Olive" title="b3">L2</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <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>  st <font color="Maroon" title="c2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c4">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <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> holds <br/><font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> implies <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:129"/><span class="lab"><font color="Green" title="E96">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:129"/><span class="lab"><font color="Green" title="E97">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:129"/><span class="lab"><font color="Green" title="E98">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L1</font> or  not <font color="Maroon" title="c2">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font> or <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:129"/><span class="lab"><font color="Green" title="E99">A4</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L1</font>
 <span class="kw">and </span><br/><a NAME="E5:129"/><span class="lab"><font color="Green" title="E100">A5</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font></span><br/>

<a NAME="E6:129"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E3:129"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="ref" href="euclidlp.html#T101" target="_self" title="EUCLIDLP:th.101">Th101</a></span>;<br/>
<a NAME="E7:129"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="ref" href="euclidlp.html#T114" target="_self" title="EUCLIDLP:th.114">Th114</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E9:129"/><span class="lab"><font color="Green" title="E101">A6</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E10:129"/><span class="lab"><font color="Green" title="E102">A7</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></span>;<br/>
<a NAME="E11:129"/><span class="lab"><font color="Green" title="E103">A8</font></span>: 
<font color="Maroon" title="c3">L1</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="ref" href="euclidlp.html#T72" target="_self" title="EUCLIDLP:th.72">Th72</a></span>;<br/>
<a NAME="E12:129"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="ref" href="euclidlp.html#T101" target="_self" title="EUCLIDLP:th.101">Th101</a></span>;<br/>
<a NAME="E13:129"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="ref" href="euclidlp.html#T114" target="_self" title="EUCLIDLP:th.114">Th114</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E15:129"/><span class="lab"><font color="Green" title="E104">A9</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E16:129"/><span class="lab"><font color="Green" title="E105">A10</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></span>;<br/>
<a NAME="E17:129"/><span class="lab"><font color="Green" title="E106">A11</font></span>: 
<font color="Maroon" title="c4">L2</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="ref" href="euclidlp.html#T72" target="_self" title="EUCLIDLP:th.72">Th72</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:129_1_1"/>
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> or <font color="Maroon" title="c7">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_1"><span class="kw">case </span></a><a NAME="E1:129_1_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:129_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="txt" href="#E3:129"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="txt" href="#E15:129"><span class="lab"><font color="Green" title="E104">A9</font></span></a>, <a class="txt" href="#E16:129"><span class="lab"><font color="Green" title="E105">A10</font></span></a>, <a class="txt" href="#E10:129"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="txt" href="#E11:129"><span class="lab"><font color="Green" title="E103">A8</font></span></a>, <a class="ref" href="euclidlp.html#T70" target="_self" title="EUCLIDLP:th.70">Th70</a>, <a class="ref" href="euclidlp.html#T113" target="_self" title="EUCLIDLP:th.113">Th113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_2"><span class="kw">case </span></a><a NAME="E1:129_1_1_2"/><span class="lab"><font color="Green" title="E107">A12</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font></span><br/><div class="add"><a NAME="E2:129_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></span>;<br/><a NAME="E3:129_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A13</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T22" title="EUCLID_4:th.22">EUCLID_4:22</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E5:129_1_1_2"/><span class="lab"><font color="Green" title="E109">A14</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">and </span><br/><a NAME="E6:129_1_1_2"/><span class="lab"><font color="Green" title="E110">A15</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:129"><span class="lab"><font color="Green" title="E106">A11</font></span></a>, <a class="ref" href="euclidlp.html#T58" target="_self" title="EUCLIDLP:th.58">Th58</a></span>;<br/><a NAME="E7:129_1_1_2"/><span class="lab"><font color="Green" title="E111">A16</font></span>: 
<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="c6">x2</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:129"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="txt" href="#E5:129_1_1_2"><span class="lab"><font color="Green" title="E109">A14</font></span></a>, <a class="txt" href="#E6:129_1_1_2"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E9:129_1_1_2"/><span class="lab"><font color="Green" title="E112">A17</font></span>: 
<font color="Maroon" title="c9">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x1</font>
 <span class="kw">and </span><br/><a NAME="E10:129_1_1_2"/><span class="lab"><font color="Green" title="E113">A18</font></span>: 
<font color="Maroon" title="c9">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:129"><span class="lab"><font color="Green" title="E103">A8</font></span></a>, <a class="ref" href="euclidlp.html#T58" target="_self" title="EUCLIDLP:th.58">Th58</a></span>;<br/><a NAME="E11:129_1_1_2"/><span class="lab"><font color="Green" title="E114">A19</font></span>: 
<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="c9">x0</font>,<font color="Maroon" title="c7">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:129"><span class="lab"><font color="Green" title="E105">A10</font></span></a>, <a class="txt" href="#E9:129_1_1_2"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E10:129_1_1_2"><span class="lab"><font color="Green" title="E113">A18</font></span></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></span>;<br/><a NAME="E12:129_1_1_2"/><span class="lab"><font color="Green" title="E115">A20</font></span>: 
<font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="txt" href="#E15:129"><span class="lab"><font color="Green" title="E104">A9</font></span></a>, <a class="txt" href="#E16:129"><span class="lab"><font color="Green" title="E105">A10</font></span></a>, <a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E101">A6</font></span></a>, <a class="txt" href="#E1:129_1_1_2"><span class="lab"><font color="Green" title="E107">A12</font></span></a>, <a class="txt" href="#E9:129_1_1_2"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E10:129_1_1_2"><span class="lab"><font color="Green" title="E113">A18</font></span></a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th79</a></span>;<br/><a NAME="E13:129_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font> <a href="rvsum_1.html#R1" title="RVSUM_1:pred.1">are_orthogonal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></span>;<br/><a NAME="E14:129_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A21</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D18" title="RVSUM_1:def.18">RVSUM_1:def 18</a></span>;<br/><a NAME="E15:129_1_1_2"/>
<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="c7">x1</font>,<font color="Maroon" title="c9">x0</font>,<font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="txt" href="#E15:129"><span class="lab"><font color="Green" title="E104">A9</font></span></a>, <a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E101">A6</font></span></a>, <a class="txt" href="#E10:129_1_1_2"><span class="lab"><font color="Green" title="E113">A18</font></span></a>, <a class="txt" href="#E12:129_1_1_2"><span class="lab"><font color="Green" title="E115">A20</font></span></a>, <a class="ref" href="euclidlp.html#T50" target="_self" title="EUCLIDLP:th.50">Th50</a>, <a class="ref" href="euclidlp.html#T97" target="_self" title="EUCLIDLP:th.97">Th97</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E17:129_1_1_2"/><span class="lab"><font color="Green" title="E117">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E18:129_1_1_2"/><span class="lab"><font color="Green" title="E118">A23</font></span>: 
<font color="Maroon" title="c8">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="c7">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="c9">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="c6">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:129"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E6:129_1_1_2"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="ref" href="euclidlp.html#T93" target="_self" title="EUCLIDLP:th.93">Th93</a></span>;<br/><a NAME="E19:129_1_1_2"/><span class="lab"><font color="Green" title="E119">A24</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:129_1_1_2"><span class="lab"><font color="Green" title="E117">A22</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E120">A25</font></span>: <font color="Maroon" title="c8">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">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="c6">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="c7">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="c9">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:129_1_1_2"><span class="lab"><font color="Green" title="E118">A23</font></span></a>, <a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">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="c6">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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:129_1_1_2"><span class="lab"><font color="Green" title="E119">A24</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<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="c6">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></span>
<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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></span>
;<br/><span class="lab"><font color="Green" title="E121">A26</font></span>: <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">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="c6">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="c7">x1</font></span>)</span></span>)</span></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>

<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">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="c6">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="c7">x1</font></span>)</span></span>)</span></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T6" title="EUCLID_4:th.6">EUCLID_4:6</a></span>
<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#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T27" title="EUCLID_4:th.27">EUCLID_4:27</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:129_1_1_2"><span class="lab"><font color="Green" title="E116">A21</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E122">A27</font></span>: <font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font> = 
<font color="Maroon" title="c9">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="c6">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></span>
<br/>.= 
<font color="Maroon" title="c9">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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T60" title="RVSUM_1:th.60">RVSUM_1:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T9" target="_self" title="EUCLIDLP:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T10" target="_self" title="EUCLIDLP:th.10">Th10</a></span>
;<br/><a NAME="E23:129_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="txt" href="#E15:129"><span class="lab"><font color="Green" title="E104">A9</font></span></a>, <a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E101">A6</font></span></a>, <a class="txt" href="#E10:129"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="txt" href="#E1:129_1_1_2"><span class="lab"><font color="Green" title="E107">A12</font></span></a>, <a class="txt" href="#E5:129_1_1_2"><span class="lab"><font color="Green" title="E109">A14</font></span></a>, <a class="txt" href="#E6:129_1_1_2"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th79</a></span>;<br/><a NAME="E24:129_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font> <a href="rvsum_1.html#R1" title="RVSUM_1:pred.1">are_orthogonal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></span>;<br/><span class="kw">then </span> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:129_1_1_2"><span class="lab"><font color="Green" title="E120">A25</font></span></a>, <a class="ref" href="rvsum_1.html#D18" title="RVSUM_1:def.18">RVSUM_1:def 18</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T25" title="EUCLID_4:th.25">EUCLID_4:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x1</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:129_1_1_2"><span class="lab"><font color="Green" title="E122">A27</font></span></a>, <a class="ref" href="euclid_4.html#T25" title="EUCLID_4:th.25">EUCLID_4:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a3</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:129_1_1_2"><span class="lab"><font color="Green" title="E121">A26</font></span></a></span>
;<br/><a NAME="E26:129_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">a1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:129_1_1_2"><span class="lab"><font color="Green" title="E108">A13</font></span></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></span>;<br/><a NAME="E27:129_1_1_2"/><span class="kw">then </span>
<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/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A28</font></span>: <font color="Maroon" title="c8">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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="c7">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="c6">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="c9">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:129_1_1_2"><span class="lab"><font color="Green" title="E120">A25</font></span></a>, <a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a></span>
<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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">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="c6">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="c6">x2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T23" target="_self" title="EUCLIDLP:th.23">Th23</a></span>
<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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></span>
<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="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></span>
;<br/><a NAME="E29:129_1_1_2"/><span class="lab"><font color="Green" title="E124">A29</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:129_1_1_2"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></span>;<br/><a NAME="E30:129_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129_1_1_2"><span class="lab"><font color="Green" title="E109">A14</font></span></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></span>;<br/><a NAME="E31:129_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">x2</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c7">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:129_1_1_2"><span class="lab"><font color="Green" title="E123">A28</font></span></a>, <a class="txt" href="#E29:129_1_1_2"><span class="lab"><font color="Green" title="E124">A29</font></span></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E32:129_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:129_1_1_2"><span class="lab"><font color="Green" title="E114">A19</font></span></a>, <a class="txt" href="#E7:129_1_1_2"><span class="lab"><font color="Green" title="E111">A16</font></span></a>, <a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:129"/>
<font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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