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<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">S'</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>

<b>assume </b><a NAME="E1:45_1_2"/><i><font color="Green">E24</font></i>: 
(  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font> st <br/>( ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> iff ( <font color="Olive">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) ) &amp;  <a href="setfam_1.html#K5">union</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> ) &amp;  ex <font color="Olive">F'</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font> st <br/>( ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F'</font> iff ( <font color="Olive">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) ) &amp;  <a href="setfam_1.html#K5">union</a> <font color="Olive">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:45_1_2"/><i><font color="Green">E25</font></i>: 
( ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) ) &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> )
 ;<br/>
<b>consider </b><font color="Maroon">F'</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E5:45_1_2"/><i><font color="Green">E27</font></i>: 
( ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> iff ( <font color="Olive">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) ) &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/><a NAME="E5:45_1_2_1_1"/>
( <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:45_1_2_1_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:45_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S'</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S'</font>
 ;<br/><b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/><a NAME="E5:45_1_2_1_2"/>
( <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:45_1_2_1_2"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:45_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S'</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:45_1_2"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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