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

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

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E20</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E23</font></i>: 
( 0 <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> 1 )
 ;<br/>

<b>consider </b><font color="Maroon">eps1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E24</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps1</font> &amp; <font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:32"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">eps1</font> = <font color="Maroon">eps1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> or <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">eps1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1"/>
<font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:32_1_1"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:32_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  )
 ;<br/><a NAME="E6:32_1_1"/>
<font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:32_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:32_1_1"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font></span>)</span> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:32_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  )
 ;<br/><b>take </b>
<font color="Maroon">r1</font>
;<br/><b>take </b>
<font color="Maroon">r2</font>
;<br/><a NAME="E11:32_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">eps1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E10:32_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E12:32_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</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">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E13:32_1_1"/>
( ( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; ( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:32_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( ( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; ( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E15:32_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:32_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:32_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E17:32_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:32_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">eps</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">eps1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">eps</font>
 ;<br/><a NAME="E19:32_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="urysohn2.html#E19:32_1_1"><i><font color="Green">E33</font></i></a>, , <a class="txt" href="urysohn2.html#E2:32"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E20:32_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E17:32_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>hence </b><a NAME="E21:32_1_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</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">r2</font> &amp; <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E15:32_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="urysohn2.html#E16:32_1_1"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, <a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">eps1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:32_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:32_1_2"/>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">eps1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:32_1_2"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">eps1</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:32_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">eps1</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  )
 ;<br/><b>take </b>
<font color="Maroon">r1</font>
;<br/><b>take </b>
<font color="Maroon">r2</font>
;<br/><a NAME="E8:32_1_2"/><i><font color="Green">E59</font></i>: 
not <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 <b>by </b><i>, <a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/><a NAME="E9:32_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:32_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 <b>by </b><i><a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/><a NAME="E11:32_1_2"/>
( ( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; ( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:32_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
( ( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; ( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:32_1_2"/>
<font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">eps1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E14:32_1_2"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</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">r2</font> &amp; <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">eps</font> )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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