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

<b>thus </b><a NAME="E1:1_1_1"/>
( <font color="Maroon">X</font> is <a href="coh_sp.html#V1">binary_complete</a> implies for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="coh_sp.html#D2">COH_SP:def 2</a><br/>

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:1_1_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:1_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_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="E1:1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:1_1_1"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>hence </b><a NAME="E3:1_1_1"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="coh_sp.html#D2">COH_SP:def 2</a><br/>


</div>
