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<b>let </b><font color="Maroon" title="c1">S</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"> ( <font color="Maroon" title="c1">S</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> implies <font color="Maroon" title="c1">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_1"/>
( <font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">S</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><b>by </b><i><a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="realset1.html#D4" target="_self" title="REALSET1:def.4">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1_1"/>
<font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">T</font> = <font color="Maroon" title="c1">S</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E3:34_1_1_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>hence </b><a NAME="E4:34_1_1_1_1"/>
<font color="Maroon" title="c1">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">S</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_1_2"/>
<font color="Maroon" title="c1">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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