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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1_2"/><i><font color="Green">E21</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="cantor_1.html#M2" target="_self">prebasis</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#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> holds <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Olive">b<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) ) )
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green">E22</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="cantor_1.html#M2" target="_self">prebasis</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> holds <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Olive">b<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="cantor_1.html#M2" target="_self">prebasis</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:27_1_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> holds <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="cantor_1.html#E1:27_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="cantor_1.html#M2" target="_self">prebasis</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:27_1_2"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> holds <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="cantor_1.html#E2:27_1_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span>;<br/><a NAME="E1:27_1_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="cantor_1.html#E4:27_1_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E2:27_1_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cantor_1.html#E6:27_1_2"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:27_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cantor_1.html#E2:27_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:27_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cantor_1.html#E2:27_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cantor_1.html#T19" target="_self">Th19</a></i>;<br/>


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