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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V5">being_T4</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="urysohn3.html#M2" target="_self">Rain</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E2:16"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a> ;<br/>

<a NAME="E1:16_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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="E2:16_1"/>
( 0 is   <a href="supinf_1.html#NM3">R_eal</a> &amp; 1 is   <a href="supinf_1.html#NM3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = 0, <font color="Maroon">c<sub>7</sub></font> = 1 as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:16_1"/><i><font color="Green">E18</font></i>: 
 <a href="urysohn1.html#K2">DYADIC</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="urysohn2.html#T32">URYSOHN2:32</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_1_1"/>
 <a href="measure5.html#K6">{}</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="measure5.html#K6">{}</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:16_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E5:16_1_1_1"/>
( not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_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="E1:16_1_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="urysohn1.html#T2">URYSOHN1:2</a></i>;<br/><a NAME="E3:16_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:16_1_1_1_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E3:16_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E1:16_1_1_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/><a NAME="E4:16_1_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><b>hence </b><a NAME="E5:16_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:16_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:16_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:16_1_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="urysohn3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E4:16_1_1_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:16_1_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E4:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E1:16_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E4:16_1_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E5:16_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#T2">URYSOHN1:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:16_1_1_2"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_2_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:16_1_1_2_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E4:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E1:16_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E4:16_1_1_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E5:16_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:16_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:16_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="urysohn3.html#E4:16_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:16_1_1_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E5:16_1_1_3"/>
( not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_3_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_3_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_3_1"><i><font color="Green">E21</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/><a NAME="E3:16_1_1_3_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn1.html#T2">URYSOHN1:2</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:16_1_1_3_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_3_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E3:16_1_1_3_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E1:16_1_1_3_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/><a NAME="E4:16_1_1_3_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><b>hence </b><a NAME="E5:16_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:16_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn1.html#T2">URYSOHN1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:16_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:16_1_1_3"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:16"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K3">DOM</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="urysohn3.html#E2:16"><i><font color="Green">E16</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K3">DOM</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:16"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>]
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:16"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  implies for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:16"><i><font color="Green">E17</font></i></a></i>;<br/>


</div>
