<?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> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><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>thus </b><a NAME="E1:43"/>
(  <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> implies  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:43_1"/><i><font color="Green">E31</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:43_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 ;<br/>

<a NAME="E3:43_1"/><i><font color="Green">E33</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E1:43_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="toprns_1.html#E2:43_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:43_1_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E1:43_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E1:43_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:43_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:43_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="toprns_1.html#E1:43_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E3:43_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E31</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 ;<br/>

<a NAME="E3:43"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E7:43"/><i><font color="Green">E36</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="toprns_1.html#E5:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 ;<br/><b>then </b> <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span></span>)</span>
<b>by </b><i><a class="txt" href="toprns_1.html#E7:43"><i><font color="Green">E36</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>

;<br/><b>hence </b><a NAME="E3:43_2"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:43"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:43"/>
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E3:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<a NAME="E10:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="toprns_1.html#E6:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E4:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div>
