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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="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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:69"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:69"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D9">WAYBEL18:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>4</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#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a>  <b>by </b><i><a class="ref" href="yellow16.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E5:69"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/>
<a NAME="E6:69"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:69"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:69_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM7">Function</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ,<a href="waybel18.html#K9">Sierpinski_Space</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ;<br/><a NAME="E2:69_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="yellow16.html#E2:69"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="yellow16.html#E5:69"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow16.html#E6:69"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:69"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <a href="waybel18.html#K9">Sierpinski_Space</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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