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<b>let </b><font color="Maroon">C1</font> be   <a href="coh_sp.html#NM1">Coherence_Space</a>, <font color="Maroon">C2</font> be   <a href="coh_sp.html#NM1">Coherence_Space</a>;<br/>

<b>thus </b><a NAME="E1:147"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C1</font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C2</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:147_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font></span>)</span>
 ;<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="E3:147_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>, <font color="Maroon">a2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E5:147_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">a2</font> &amp; ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a>, </i>;<br/>
<a NAME="E6:147_1"/>
( <font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">a2</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E7:147_1"/><b>then </b>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C1</font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="cohsp_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E8:147_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C1</font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C2</font></span>)</span>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">z</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="E2:147"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C1</font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C2</font></span>)</span>
 ;<br/>

<a NAME="E3:147"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C1</font> ) or ( <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C2</font> ) ) )
 
<b>by </b><i><a class="ref" href="cohsp_1.html#D14" target="_self">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:147_2"/>
( ( <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C1</font> ) or ( <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C2</font> ) )
 </span><b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cohsp_1.html#D14" target="_self">Def14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_2_1"><b>suppose </b></a><a NAME="E1:147_2_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:147_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> <b>by </b><i><a class="ref" href="coh_sp.html#T1">COH_SP:1</a></i>;<br/><a NAME="E6:147_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font> )
 <b>by </b><i>, , , <a class="ref" href="cohsp_1.html#D15" target="_self">Def15</a>, </i>;<br/><b>hence </b><a NAME="E7:147_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font></span>)</span>
 <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="hsNdiv(this)" href="javascript:()" title="147_2_2"><b>suppose </b></a><a NAME="E1:147_2_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:147_2_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> <b>by </b><i><a class="ref" href="coh_sp.html#T1">COH_SP:1</a></i>;<br/><a NAME="E6:147_2_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font> )
 <b>by </b><i>, , , <a class="ref" href="cohsp_1.html#D16" target="_self">Def16</a>, </i>;<br/><b>hence </b><a NAME="E7:147_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">C2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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