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<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="E1:19"/><i><font color="Green">E17</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>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E2:19"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:19"/><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#K3">DOM</a> 
 ;<br/>

<a NAME="E4:19"/><i><font color="Green">E20</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="txt" href="urysohn3.html#E3:19"><i><font color="Green">E19</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_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#E4:19"><i><font color="Green">E20</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="19_1_1"><b>suppose </b></a><a NAME="E1:19_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="E2:19_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn3.html#E2:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E3:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:19_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a></i>;<br/><b>hence </b><a NAME="E4:19_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/><i><font color="Green">E21</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>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:19_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#K1">dyadic</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/><a NAME="E4:19_1_2"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</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="Maroon">c<sub>7</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#E1:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn3.html#E3:19"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E1:19_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="urysohn3.html#E3:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E6:19_1_2"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>7</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E3:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#T5">URYSOHN1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><b>case </b></a><a NAME="E1:19_1_2_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="urysohn1.html#T12">URYSOHN1:12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:19_1_2_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="urysohn1.html#T12">URYSOHN1:12</a></i>;<br/><a NAME="E4:19_1_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn3.html#E2:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E3:19"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E1:19_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:19_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn3.html#E2:19_1_2_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><b>case </b></a><a NAME="E1:19_1_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_2"/><b>then </b>
( 0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>7</sub></font> &amp; 0 <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#T12">URYSOHN1:12</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E3:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn3.html#E4:19_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn3.html#E6:19_1_2"><i><font color="Green">E24</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="E8:19_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3"><b>suppose </b></a><a NAME="E1:19_1_3"/>
<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="E2:19_1_3"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</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>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn3.html#E2:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E3:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:19_1_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><b>hence </b><a NAME="E4:19_1_3"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:19_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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