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<b>let </b><font color="Maroon">T</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">A</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">G1</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E4:2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E5:2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E6:2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/>
<b>then consider </b><font color="Maroon">G0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G0</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, , <a class="ref" href="urysohn1.html#T31">URYSOHN1:31</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> ) );<br/>
<a NAME="E9:2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6">URYSOHN1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">G0</font>
;<br/><b>thus </b><a NAME="E2:2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">G0</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">G1</font>
;<br/><b>thus </b><a NAME="E2:2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">G1</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:2"/>
 ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1">dyadic</a> 0, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0 holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>then consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1">dyadic</a> 0, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0 holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">x</font>]
 ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E13:2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<a NAME="E1:2_2"/>
( 0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6">URYSOHN1:6</a></i>;<br/>
<b>hence </b><a NAME="E2:2_2"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:2"/>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</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">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0, <font color="Maroon">r2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0;<br/>

<b>assume </b><a NAME="E1:2_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/>

<a NAME="E2:2_3"/><i><font color="Green">E47</font></i>: 
( ( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6">URYSOHN1:6</a></i>;<br/>
<a NAME="E3:2_3"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:2_3"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Maroon">r2</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">F</font> <a href="urysohn1.html#K4">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , , , <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:2"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> 0  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</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">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> ) ) )
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>


</div>
