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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="waybel18.html#V1" title="WAYBEL18:attr.1">TopSpace-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">F</font>  holds not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <font color="Maroon" title="c2">J</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">G</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">xi</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Maroon" title="c3">i</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">G</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">F</font>;<br/>



<b>assume </b><a NAME="E2:23"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">xi</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 <b>and </b><br/><a NAME="E6:23"/><i><font color="Green" title="E26">A4</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">xi</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="yellow17.html#T21" target="_self" title="YELLOW17:th.21">Th21</a></i>;<br/>
<a NAME="E7:23"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <font color="Maroon" title="c2">J</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <font color="Maroon" title="c2">J</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">G1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">F</font> <b>by </b><i><a class="txt" href="yellow17.html#E5:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:23_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <font color="Maroon" title="c2">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:23_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow17.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">Ai</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Maroon" title="c3">i</font></span>)</span><b> such that </b><br/><a NAME="E9:23"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c8">Ai</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
 <b>and </b><br/><a NAME="E10:23"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c5">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Ai</font>
 <b>and </b><br/><a NAME="E11:23"/><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c8">Ai</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E12:23"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">Ai</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E4:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="yellow17.html#E6:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="yellow17.html#T18" target="_self" title="YELLOW17:th.18">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">Ai</font>
;<br/>

<b>thus </b><a NAME="E13:23"/>
<font color="Maroon" title="c8">Ai</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow17.html#E9:23"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:23"/>
<font color="Maroon" title="c5">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Ai</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E10:23"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/>

<b>thus </b><a NAME="E15:23"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">Ai</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E5:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="yellow17.html#E12:23"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>

<b>thus </b><a NAME="E16:23"/>
<font color="Maroon" title="c8">Ai</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E11:23"><i><font color="Green" title="E29">A8</font></i></a></i>;<br/>


</div>
