<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L0</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>  st <font color="Olive" title="b1">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b2">L1</font> &amp; <font color="Olive" title="b2">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Olive" title="b3">L2</font> holds <br/><font color="Olive" title="b1">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b3">L2</font></span><br/><b>let </b><font color="Maroon" title="c2">L0</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L1</font> &amp; <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font> implies <font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font> )</span><br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E58">A1</font></i>: 
( <font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L1</font> &amp; <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">x0</font>, <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">x3</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="E3:69"/><i><font color="Green" title="E59">A2</font></i>: 
( <font color="Maroon" title="c2">L0</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="c5">x0</font>,<font color="Maroon" title="c6">x1</font> &amp; <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="c7">x2</font>,<font color="Maroon" title="c8">x3</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c8">x3</font> <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#D8" target="_self" title="EUCLIDLP:def.8">Def8</a></i>;<br/>
<a NAME="E4:69"/><i><font color="Green" title="E60">A3</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c8">x3</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="#E3:69"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y0</font>, <font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font>, <font color="Maroon" title="c12">y3</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="E6:69"/><i><font color="Green" title="E61">A4</font></i>: 
( <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">y0</font>,<font color="Maroon" title="c10">y1</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="c11">y2</font>,<font color="Maroon" title="c12">y3</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</font> )
 <b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/>
<a NAME="E7:69"/><i><font color="Green" title="E62">A5</font></i>: 
( <font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</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="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</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="#E6:69"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E8:69"/>
( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c9">y0</font>,<font color="Maroon" title="c10">y1</font> &amp; <font color="Maroon" title="c8">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c9">y0</font>,<font color="Maroon" title="c10">y1</font> )
 
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E6:69"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:69"/><i><font color="Green" title="E63">A6</font></i>: 
<font color="Maroon" title="c8">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></i>;<br/>
<a NAME="E11:69"/>
<font color="Maroon" title="c8">x3</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="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</font>
 
<b>by </b><i><a class="txt" href="#E4:69"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="#E7:69"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="#E10:69"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E12:69"/><b>then </b>
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</font>
 
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E6:69"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self" title="EUCLIDLP:th.38">Th38</a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th49</a></i>;<br/>
<b>hence </b><a NAME="E13:69"/>
<font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E6:69"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="euclidlp.html#D8" target="_self" title="EUCLIDLP:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
