<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">xi</font> be   <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="Maroon">i</font></span>)</span>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K2">product_prebasis</a> <font color="Maroon">J</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E20</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi</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="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel18.html#K2">product_prebasis</a> <font color="Maroon">J</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> not <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"><font color="Maroon">xi</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:21"/>
<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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <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> ;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f'</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">xi</font>;<br/>
<a NAME="E5:21"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f'</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <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"><font color="Maroon">xi</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Ag</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:21"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f'</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ag</font>
 <b>and </b><br/><a NAME="E8:21"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Ag</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">Ai2</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">i2</font></span>)</span><b> such that </b><br/><a NAME="E10:21"/>
<font color="Maroon">Ai2</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E11:21"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i2</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Ai2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ag</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E5:21"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<a NAME="E12:21"/><i><font color="Green">E31</font></i>: 
not <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"><font color="Maroon">xi</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i2</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E2:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow17.html#E5:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow17.html#E7:21"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E13:21"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Ai2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4">.</a> <font color="Maroon">i2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">Ai2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4">.</a> <font color="Maroon">i2</font></span>)</span>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i2</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Ai2</font> = 
 <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/>
<br/>.= 
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>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<b>hence </b><a NAME="E3:21_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow17.html#E2:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow17.html#E5:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow17.html#E7:21"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:21"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 ;<br/>

<a NAME="E2:21_2"/><b>then </b><i><font color="Green">E45</font></i>: 
not <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E8:21"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow17.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Ai2'</font> = <font color="Maroon">Ai2</font> as   <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>by </b><i/>;<br/>
<a NAME="E4:21_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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">Ai2'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow17.html#T2" target="_self">Th2</a>, , <a class="txt" href="yellow17.html#E7:21"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:21_2"/><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"><font color="Maroon">xi</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <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">Ai2'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow17.html#E11:21"><i><font color="Green">E30</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:21"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i2</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i>, <a class="txt" href="yellow17.html#E7:21"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E16:21"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E5:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow17.html#E7:21"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
