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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> iff  not <font color="Maroon" title="c1">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">z</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">thus </span><a NAME="E1:4"/>
( 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> implies  not <font color="Maroon" title="c1">X</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  implies 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:4_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c1">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:4_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E4:4_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E5:4_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:4_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c1">X</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="c5">z</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c3">x</font> = 
<font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:4_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E2:4_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:4_1"/>
 not <font color="Maroon" title="c1">X</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E5:4_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T131" title="ZFMISC_1:th.131">ZFMISC_1:131</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:4"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 not <font color="Maroon" title="c1">X</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="firebrick">::  thesis: </font></span></a><span class="hide"> 2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E3:4"/><span class="kw">then </span>
( <font color="Maroon" title="c1">X</font> <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"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c1">X</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"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:4"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E6:4"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 not <font color="Maroon" title="c3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">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="#E2:4"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E7:4"/>
<font color="Maroon" title="c3">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:4"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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