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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:53"/>
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:53"/><b>then </b>
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E48</font></i>: 
(  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E6:53"/>
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:53"><i><font color="Green">E48</font></i></a>, <a class="ref" href="euclid.html#T52">EUCLID:52</a></i>;<br/>
<a NAME="E7:53"/><b>then </b>
 <a href="euclid.html#K19">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E8:53"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
(  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:53"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E3:53_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E4:53_1_1"/><b>then </b>
1 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:53_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E5:53_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
1 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 0
 ;<br/><a NAME="E3:53_1_2"/><b>then </b>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E4:53_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:53"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><b>hence </b><a NAME="E5:53_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
