<?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">x0</font>, <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">y0</font>, <font color="Maroon" title="c5">y1</font> be    <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>;<br/><b>let </b><font color="Maroon" title="c6">L1</font>, <font color="Maroon" title="c7">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>assume </b><a NAME="E1:86"/><i><font color="Green" title="E73">A1</font></i>: 
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L1</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L1</font> &amp; <font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c4">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L2</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L2</font> &amp; <font color="Maroon" title="c4">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c6">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c7">L2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">x3</font>, <font color="Maroon" title="c10">y2</font>, <font color="Maroon" title="c11">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="E3:86"/><i><font color="Green" title="E74">A2</font></i>: 
( <font color="Maroon" title="c6">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">x2</font>,<font color="Maroon" title="c9">x3</font> &amp; <font color="Maroon" title="c7">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="c10">y2</font>,<font color="Maroon" title="c11">y3</font> &amp; <font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:86"/><i><font color="Green" title="E75">A3</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:86"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:86"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:86"/><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:86"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:86"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:86"/><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:86"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E10:86"/><i><font color="Green" title="E78">A6</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">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="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">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> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:86"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>
<a NAME="E11:86"/><b>then </b><i><font color="Green" title="E79">A7</font></i>: 
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E5:86"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="euclidlp.html#E7:86"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>;<br/>
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E5:86"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="euclidlp.html#E9:86"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E11:86"><i><font color="Green" title="E79">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c13">t</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E7:86"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E13:86"/>
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:86"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>


</div>
