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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A1</font> <a href="setlim_2.html#K3" title="SETLIM_2:func.3">(\)</a> <font color="Olive" title="b2">A2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K3" title="SETLIM_2:func.3">(\)</a> <font color="Maroon" title="c3">A2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c3">A2</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K3" title="SETLIM_2:func.3">(\)</a> <font color="Maroon" title="c3">A2</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:11"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c3">A2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K3" title="SETLIM_2:func.3">(\)</a> <font color="Maroon" title="c3">A2</font></span>)</span></span><br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c2">A1</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c3">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">n1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n1</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<a NAME="E5:11"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A2</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n1</font>
 
<b>by </b><i><a class="txt" href="#E2:11"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A2</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K3" title="SETLIM_2:func.3">(\)</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n1</font>
 
<b>by </b><i><a class="ref" href="setlim_2.html#D3" target="_self" title="SETLIM_2:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E8:11"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K3" title="SETLIM_2:func.3">(\)</a> <font color="Maroon" title="c3">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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