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<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel18.html#V1">TopSpace-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="waybel18.html#K4">.</a> <font color="Olive">i</font> is <a href="compts_1.html#V2">compact</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> =  <a href="waybel18.html#K2">product_prebasis</a> <font color="Maroon">J</font> as    <a href="cantor_1.html#M2">prebasis</a> of  <a href="waybel18.html#K3">product</a> <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="waybel18.html#D3">WAYBEL18:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:25"/>
not  <a href="waybel18.html#K3">product</a> <font color="Maroon">J</font> is <a href="compts_1.html#V2">compact</a>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:25"/><i><font color="Green">E22</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:25"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font>  holds not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">G</font>
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>] <b>means </b>for <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font>  holds not <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">G</font>;<br/>
<a NAME="E7:25"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  ex <font color="Olive">xi</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4">.</a> <font color="Olive">i</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">xi</font>,<font color="Olive">i</font>]
 
<b>by </b><i>, <a class="txt" href="yellow17.html#E7:25"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow17.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">J</font></span>)</span><b> such that </b><br/><a NAME="E9:25"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="waybel18.html#K5">.</a> <font color="Olive">i</font>,<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="yellow17.html#S1" target="_self">YELLOW17:sch 1</a>(<a class="txt" href="yellow17.html#E9:25"><i><font color="Green">E27</font></i></a>);<br/></i>
<a NAME="E10:25"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">J</font></span>)</span>
 
;<br/>
<a NAME="E11:25"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:25"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E14:25"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E6:25"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">Ai</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E16:25"/>
<font color="Maroon">Ai</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E17:25"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Ai</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E13:25"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font> <b>by </b><i><a class="txt" href="yellow17.html#E13:25"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E19:25"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font>
 
<b>by </b><i>, <a class="txt" href="yellow17.html#E14:25"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E20:25"/><b>then </b>
<font color="Maroon">f</font> <a href="waybel18.html#K5">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:25"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel18.html#K5">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ai</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E22:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel18.html#K5">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E14:25"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E23:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel18.html#K5">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<b>hence </b><a NAME="E24:25"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow17.html#T3" target="_self">Th3</a></i>;<br/>


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