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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-alphabet</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">  for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">u</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">t</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> holds <br/><font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font>, <font color="Maroon" title="c3">u</font> be   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>; <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="c2">t</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c3">u</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c2">t</font> implies <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">R</font> =  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E1:67"/>
 the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D32" target="_self" onmouseover="rs('qc_lang1/D32')" onmouseout="rh()">Def32</a></span>;<br/>
<a NAME="E2:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A1</font></span>: 
 the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D5" onmouseover="rs('wellord1/D5')" onmouseout="rh()">WELLORD1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:67"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
( <font color="Maroon" title="c2">t</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c3">u</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c2">t</font> )
 ; <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="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1"/>
( ( <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c3">u</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">t</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="c3">u</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="67_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
( <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c3">u</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"> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:67_1_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
( <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c8">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:67_1_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</a></span>;<br/><span class="kw">thus </span><a NAME="E6:67_1_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E5:67_1_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_2"/>
(  not <font color="Maroon" title="c2">t</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="c3">u</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"> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_2_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c2">t</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="c3">u</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="67_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_2_1_1"/>
 not <font color="Maroon" title="c2">t</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"> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font></span><br/><div class="add"><a NAME="E2:67_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A6</font></span>: 
(  not <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not <font color="Maroon" title="c3">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A2</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="E3:67_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
( <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp; <font color="Maroon" title="c3">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:67_1_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c3">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E2:67_1_2_1_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:67_1_2_1_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E3:67_1_2_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="relat_2.html#D4" onmouseover="rs('relat_2/D4')" onmouseout="rh()">RELAT_2:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_2_1_2"/>
 not <font color="Maroon" title="c3">u</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"> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font></span><br/><div class="add"><a NAME="E2:67_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A8</font></span>: 
(  not <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  not <font color="Maroon" title="c3">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A2</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="E3:67_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A9</font></span>: 
( <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp; <font color="Maroon" title="c3">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:67_1_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c3">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E2:67_1_2_1_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:67_1_2_1_2"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E3:67_1_2_1_2"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="relat_2.html#D4" onmouseover="rs('relat_2/D4')" onmouseout="rh()">RELAT_2:def 4</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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