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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>;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> ) or ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><a NAME="E1:26_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_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="E1:26_1_1_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1"><b>case </b></a><a NAME="E1:26_1_1_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c</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="E3:26_1_1_1_1_1"/><i><font color="Green">E25</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> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:26_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</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/><a NAME="E5:26_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_1_1_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/><a NAME="E6:26_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c</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> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:26_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 <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/><b>hence </b><a NAME="E8:26_1_1_1_1_1"/>
 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">a</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">b</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2"><b>case </b></a><a NAME="E1:26_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">a1</font> = 0;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:26_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; 0 <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> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:26_1_1_1_1_2"/>
<font color="Maroon">c</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> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:26_1_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:26_1_1_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 <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="E6:26_1_1_1_1_2"/>
( <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> )
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E1:26_1_1_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_1_1_2"/>
 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">a</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">b</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_1_1_1_2"><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="E3:26_1_1"/>
 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">a</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">b</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:26_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:26_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">a1</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">b1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:26_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c</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> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">c</font>
;<br/><b>thus </b><a NAME="E7:26_1_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <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> )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3"><b>suppose </b></a><a NAME="E1:26_1_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:26_1_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <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> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:26_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c</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> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">c</font>
;<br/><b>thus </b><a NAME="E5:26_1_3"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <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> )
 <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4"><b>suppose </b></a><a NAME="E1:26_1_4"/><i><font color="Green">E55</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_4_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4_1_1_1"><b>case </b></a><a NAME="E1:26_1_4_1_1_1"/><i><font color="Green">E58</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c</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="E3:26_1_4_1_1_1"/><i><font color="Green">E59</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> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</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/><a NAME="E5:26_1_4_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1
 ;<br/><a NAME="E7:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c</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> <span class="p1">(<span class="default"><a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 <b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E10:26_1_4_1_1_1"/>
 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">a</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">b</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4_1_1_2"><b>case </b></a><a NAME="E1:26_1_4_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">b1</font> = 1;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:26_1_4_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <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> 1 )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:26_1_4_1_1_2"/>
<font color="Maroon">c</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> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:26_1_4_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 <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="E6:26_1_4_1_1_2"/>
( <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> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:26_1_4_1_1_2"/>
 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">a</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">b</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_4"/>
 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">a</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">b</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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