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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>





<b>thus </b><a NAME="E1:41"/>
 <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</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="41_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="E1:41_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<a NAME="E2:41_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> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:41_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E3:41_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="setlim_2.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E4:41_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">k</font> )
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">k</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1"><b>suppose </b></a><a NAME="E1:41_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_2"><b>suppose </b></a><a NAME="E1:41_1_2_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">k</font>
 ;<br/><div class="add"><a NAME="E2:41_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><b>hence </b><a NAME="E3:41_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<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="E2:41"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1"><b>suppose </b></a><a NAME="E1:41_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:41_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:41_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:41_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_2"><b>suppose </b></a><a NAME="E1:41_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:41_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><a NAME="E4:41_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_2.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:41_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:41_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
