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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E2:11"/><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green">E13</font></i>: 
 <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
not  <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
 <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn2.html#T33">URYSOHN2:33</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="urysohn3.html#E4:11"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E6:11"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="urysohn3.html#E2:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="urysohn3.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:11"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:11_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:11_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E4:11_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="urysohn3.html#E1:11_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="urysohn3.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:11_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:11"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="urysohn3.html#E6:11"><i><font color="Green">E14</font></i></a>, <a class="txt" href="urysohn3.html#E7:11"><i><font color="Green">E15</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:11"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E11:11"/>
 <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:11"><i><font color="Green">E13</font></i></a>, <a class="txt" href="urysohn3.html#E10:11"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:11"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E14:11"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E8:11"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:11"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="urysohn3.html#E10:11"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn3.html#E13:11"><i><font color="Green">E18</font></i></a></i>;<br/>


</div>
