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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   <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#K1" title="SETLIM_2:func.1">(/\)</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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#K9" title="SUBSET_1:func.9">/\</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></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">  <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#K1" title="SETLIM_2:func.1">(/\)</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K9" title="SUBSET_1:func.9">/\</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></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>  <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#K1" title="SETLIM_2:func.1">(/\)</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> <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#K9" title="SUBSET_1:func.9">/\</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> )</span><br/>







<b>assume </b><a NAME="E1:9"/>
<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#K1" title="SETLIM_2:func.1">(/\)</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> <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#K9" title="SUBSET_1:func.9">/\</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></span><br/>

<b>then consider </b><font color="Maroon" title="c5">n</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="E3:9"/><i><font color="Green" title="E13">A1</font></i>: 
<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#K1" title="SETLIM_2:func.1">(/\)</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">n</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="E4:9"/>
<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">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:9"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="setlim_2.html#D1" target="_self" title="SETLIM_2:def.1">Def1</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
( <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">n</font> &amp; <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">n</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:9"/><b>then </b>
( <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; <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="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
<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#K9" title="SUBSET_1:func.9">/\</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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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