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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">A</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  iff  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a2</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a2</font> ) implies  not <font color="Maroon" title="c1">A</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
not <font color="Maroon" title="c1">A</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:30_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:30_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
( <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">a1</font> = <font color="Maroon" title="c2">a1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a2</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c5">a2</font> = <font color="Maroon" title="c3">a2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a2</font> )</span><br/><span class="kw">thus </span><a NAME="E7:30_1"/>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E3:30"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E4:30"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:30"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">A</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/>

<span class="kw">thus </span><a NAME="E6:30"/>
not <font color="Maroon" title="c1">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="realset1.html#D4" target="_self" title="REALSET1:def.4">REALSET1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E8:30"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E9:30"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/>
<a NAME="E10:30"/><span class="kw">then </span>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T26" title="ZFMISC_1:th.26">ZFMISC_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:30"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T26" title="ZFMISC_1:th.26">ZFMISC_1:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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