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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> )</span><br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span><br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">c</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23" title="XREAL_1:th.23">XREAL_1:23</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:31"/><b>then </b>
( <font color="Maroon" title="c3">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23" title="XREAL_1:th.23">XREAL_1:23</a></i>;<br/>
<a NAME="E6:31"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E7:31"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E8:31"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E29">A6</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1"/><i><font color="Green" title="E30">A7</font></i>: 
( <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> )</span><br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_1"/>
( <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:31_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_2"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:31_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:31_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_2"/>
( <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_2_1"/>
( <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E2:31_1"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1"><b>case </b></a><a NAME="E1:31_1_2_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:31_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:31"><i><font color="Green" title="E25">A2</font></i></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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_2"><b>case </b></a><a NAME="E1:31_1_2_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:31_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:31"><i><font color="Green" title="E27">A4</font></i></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><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:31"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E7:31"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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