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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <font color="Maroon">x2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E2:27"/><b>then </b>
 <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E5:27"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E6:27"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E7:27"/>
 <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:27"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:27"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E11:27"/>
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x2</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x1</font>,<font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, </i>;<br/>


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