<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">e</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span> ;<br/>
<a NAME="E1:71_1_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span>  or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:71_1_1_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Maroon" title="c5">t</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E4:71_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E2:71_1_1"/>
 ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">t</font> where <font color="Olive" title="b2">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b2">t</font> </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:71_1_1_2_1"/>
( <font color="Maroon" title="c2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  or  not <font color="Maroon" title="c2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_2_1_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">t</font> where <font color="Olive" title="b2">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b2">t</font> </span> } </span> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:71_1_1_2_1_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E5:71_1_1_2_1_1"/>
 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T3" target="_self" onmouseover="rs('qc_lang1/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E7:71_1_1_2_1_1"/>
(  not <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" onmouseover="rs('nat_1/T19')" onmouseout="rh()">NAT_1:19</a></span>;<br/><a NAME="E8:71_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_2_1_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</a>, <a class="txt" href="#E1:71_1_1_2_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a></span>;<br/><a NAME="E9:71_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E10:71_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E11:71_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">t</font> where <font color="Olive" title="b2">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b2">t</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
 not <font color="Maroon" title="c2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">t</font> where <font color="Olive" title="b2">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b2">t</font> </span> } </span> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  as   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T3" target="_self" onmouseover="rs('qc_lang1/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E3:71_1_1_2_1_2"/>
(  not <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_2_1_2"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</a></span>;<br/><a NAME="E4:71_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Maroon" title="c4">a</font>
 ;<br/><a NAME="E5:71_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E6:71_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">t</font> where <font color="Olive" title="b2">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b2">t</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:71_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon" title="c2">s</font> <a href="qc_lang1.html#R2" title="QC_LANG1:pred.2">&lt;</a> <font color="Olive" title="b1">t</font> </span> } </span>  is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
