<?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">q1</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> holds <br/> <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b4">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font></span><br/><b>let </b><font color="Maroon" title="c2">q1</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> implies  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )</span><br/>









<b>consider </b><font color="Maroon" title="c6">x1</font>, <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="E2:53"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E60"><i><font color="Green" title="E29">Lm8</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:53"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</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="E5:53"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q1</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c5">q2</font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E60"><i><font color="Green" title="E29">Lm8</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:53"/>
 <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="euclid_4.html#E2:53"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="euclid_4.html#E3:53"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="euclid_4.html#E5:53"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="euclid_4.html#T11" target="_self" title="EUCLID_4:th.11">Th11</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>
