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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E3:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:31"/><b>then </b>
( <font color="Maroon">c</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/>
<a NAME="E6:31"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E7:31"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E7:31"><i><font color="Green">E26</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E8:31"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">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">E27</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:31_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">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="hs2(this)" href="javascript:()" title="31_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_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_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_1_1"/>
( <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_1_2"/>
not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_1_2"/>
( <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_1"/>
( <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>


</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">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> )
 </span><b>by </b><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">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 ;<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#E4:31"><i><font color="Green">E25</font></i></a></i>;<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">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:31_1_2_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font>
 ;<br/><b>hence </b><a NAME="E3:31_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:31_1"><i><font color="Green">E27</font></i></a></i>;<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
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:31"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div>
