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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> ) &amp; <font color="Olive" title="b1">X</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font></span><br/><b>let </b><font color="Maroon" title="c2">X</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"> ( ( for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> ) &amp; <font color="Maroon" title="c2">X</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> implies  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font></span><br/>

<b>consider </b><font color="Maroon" title="c3">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:63"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c3">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> )
 <b>and </b><br/><a NAME="E5:63"/><i><font color="Green" title="E33">A4</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; ( for <font color="Olive" title="b3">Z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Olive" title="b2">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z2</font> ) )
 <b>by </b><i><a class="txt" href="card_3.html#E2:63"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="card_3.html#T61" target="_self" title="CARD_3:th.61">Th61</a></i>;<br/>
<div><i><font color="Green" title="E34">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">Z1</font>, <font color="Maroon" title="c5">Z2</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="c4">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c5">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; not <font color="Maroon" title="c4">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z2</font> implies <font color="Maroon" title="c5">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z1</font> )</span><br/><b>assume </b><a NAME="E1:63_1"/>
( <font color="Maroon" title="c4">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c5">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z2</font> or <font color="Maroon" title="c5">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z1</font> )</span><br/><a NAME="E2:63_1"/><b>then </b>
<font color="Maroon" title="c4">Z1</font>,<font color="Maroon" title="c5">Z2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="card_3.html#E2:63"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="card_3.html#E4:63"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:63_1"/>
( <font color="Maroon" title="c4">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z2</font> or <font color="Maroon" title="c5">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Z1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>; <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>
<b>consider </b><font color="Maroon" title="c4">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E8:63"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c4">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T26" title="WELLORD2:th.26">WELLORD2:26</a></i>;<br/>
<a NAME="E9:63"/><i><font color="Green" title="E36">A7</font></i>: 
( <font color="Maroon" title="c4">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">R</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E8:63"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  ex <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> &amp; not $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) or ( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> ) );<br/>
<b>consider </b><font color="Maroon" title="c5">Q</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E11:63"/><i><font color="Green" title="E37">A8</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1" title="RELAT_1:sch.1">RELAT_1:sch 1</a>();<br/></i>
<a NAME="E12:63"/><i><font color="Green" title="E38">A9</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<b>assume </b><a NAME="E1:63_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E2:63_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E67"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:63_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a></i>; <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>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> )</span><br/>

<b>assume </b><a NAME="E2:63_2"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font></span><br/>

<a NAME="E3:63_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> &amp; ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) )
 
<b>by </b><i><a class="txt" href="card_3.html#E9:63"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E4:63_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E2:63_2"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:63_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>; <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>
<a NAME="E13:63"/>
<font color="Maroon" title="c5">Q</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_3"/>
<font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D4" title="WELLORD1:def.4">WELLORD1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a>,<a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/>

<b>assume </b><a NAME="E1:63_3_1"/><i><font color="Green" title="E39">A11</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font></span><br/>

<a NAME="E2:63_3_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> &amp; ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) )
 
<b>by </b><i><a class="txt" href="card_3.html#E9:63"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:63_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_1"><i><font color="Green" title="E39">A11</font></i></a></i>; <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>

<b>thus </b><a NAME="E2:63_3"/>
<font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a>,<a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/>





<b>assume </b><a NAME="E1:63_3_2"/><i><font color="Green" title="E39">A12</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font></span><br/>

<div><i><font color="Green" title="E40">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_3_2_1"/><i><font color="Green" title="E41">A14</font></i>: 
( <font color="Maroon" title="c9">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">Z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z2</font> &amp; not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z2</font> ) implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/><b>given </b><font color="Maroon" title="c10">Z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E4:63_3_2_1"/><i><font color="Green" title="E42">A15</font></i>: 
( <font color="Maroon" title="c10">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z2</font> &amp; not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font></span><br/><a NAME="E5:63_3_2_1"/>
( <font color="Maroon" title="c9">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z2</font> or <font color="Maroon" title="c10">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:63"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="card_3.html#E4:63_3_2_1"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E6:63_3_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="card_3.html#E4:63_3_2_1"><i><font color="Green" title="E42">A15</font></i></a></i>; <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><i><font color="Green" title="E41">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_3_2_2"/><i><font color="Green" title="E42">A17</font></i>: 
( <font color="Maroon" title="c9">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/><b>assume </b><a NAME="E3:63_3_2_2"/>
( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font></span><br/><a NAME="E4:63_3_2_2"/><b>then </b>
not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font>
 <b>by </b><i><a class="txt" href="card_3.html#E2:63_3_2_2"><i><font color="Green" title="E42">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E5:63_3_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_2_2"><i><font color="Green" title="E42">A17</font></i></a></i>; <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><i><font color="Green" title="E42">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_3_2_3"/><i><font color="Green" title="E43">A19</font></i>: 
( <font color="Maroon" title="c9">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> &amp; not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/><b>assume </b><a NAME="E3:63_3_2_3"/>
( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font></span><br/><a NAME="E4:63_3_2_3"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font>
 <b>by </b><i><a class="txt" href="card_3.html#E2:63_3_2_3"><i><font color="Green" title="E43">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E5:63_3_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_2_3"><i><font color="Green" title="E43">A19</font></i></a></i>; <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="hs2(this)" href="javascript:()" title="63_3_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_2_4"/><i><font color="Green" title="E43">A20</font></i>: 
( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> &amp; ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font></span><br/><a NAME="E2:63_3_2_4"/><b>then </b><i><font color="Green" title="E44">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 <b>by </b><i><a class="txt" href="card_3.html#E9:63"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">Z</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="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> implies ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> iff <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> ) )</span><br/><b>assume </b><a NAME="E1:63_3_2_4_1"/>
<font color="Maroon" title="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> iff <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> )</span><br/><a NAME="E2:63_3_2_4_1"/><b>then </b>
( ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> implies <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> ) &amp; ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> ) &amp; ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> ) &amp; ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> implies <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> ) )
 <b>by </b><i><a class="txt" href="card_3.html#E1:63_3_2_4"><i><font color="Green" title="E43">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_3_2_4_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> iff <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> )
 ; <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><b>hence </b><a NAME="E4:63_3_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_2_4"><i><font color="Green" title="E44">A21</font></i></a></i>; <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>
<b>hence </b><a NAME="E6:63_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="card_3.html#E3:63_3_2"><i><font color="Green" title="E41">A16</font></i></a>, <a class="txt" href="card_3.html#E4:63_3_2"><i><font color="Green" title="E42">A18</font></i></a></i>; <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>

<b>thus </b><a NAME="E3:63_3"/>
<font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a>,<a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c7">y</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/>



<b>assume </b><a NAME="E1:63_3_3"/><i><font color="Green" title="E39">A22</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E2:63_3_3"/><b>then </b><i><font color="Green" title="E40">A23</font></i>: 
( (  ex <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) or ( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> ) ) &amp; (  ex <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) or ( ( for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> ) ) )
 
<b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_3_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c8">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_3_3_1"/><i><font color="Green" title="E41">A24</font></i>: 
( <font color="Maroon" title="c8">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z1</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">Z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z2</font> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z2</font> ) implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><b>given </b><font color="Maroon" title="c9">Z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E4:63_3_3_1"/><i><font color="Green" title="E42">A25</font></i>: 
( <font color="Maroon" title="c9">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z2</font> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/><a NAME="E5:63_3_3_1"/>
( <font color="Maroon" title="c8">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z2</font> or <font color="Maroon" title="c9">Z2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Z1</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:63"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_3_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="card_3.html#E4:63_3_3_1"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E6:63_3_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="card_3.html#E2:63_3_3_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="card_3.html#E4:63_3_3_1"><i><font color="Green" title="E42">A25</font></i></a></i>; <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>
<b>hence </b><a NAME="E4:63_3_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="card_3.html#E9:63"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_3"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_3"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>; <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>

<b>thus </b><a NAME="E4:63_3"/>
<font color="Maroon" title="c5">Q</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> is <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a>,<a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c7">y</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/>



<b>assume </b><a NAME="E1:63_3_4"/><i><font color="Green" title="E39">A26</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_4_1"/><i><font color="Green" title="E40">A27</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> iff <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )</span><br/><a NAME="E2:63_3_4_1"/><b>then </b>
( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> ) &amp; ( for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> holds <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> ) ) )
 <b>by </b><i><a class="txt" href="card_3.html#E9:63"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_4"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><b>hence </b><a NAME="E3:63_3_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_4"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_4_1"><i><font color="Green" title="E40">A27</font></i></a></i>; <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>
<b>hence </b><a NAME="E3:63_3_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_4"><i><font color="Green" title="E39">A26</font></i></a></i>; <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>

<b>thus </b><a NAME="E5:63_3"/>
<font color="Maroon" title="c5">Q</font> is <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D2" title="WELLORD1:def.2">WELLORD1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:63_3_5"/><i><font color="Green" title="E39">A28</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Y</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>consider </b><font color="Maroon" title="c7">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:63_3_5"/><i><font color="Green" title="E40">A29</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:63_3_5"/><i><font color="Green" title="E41">A30</font></i>: 
<font color="Maroon" title="c7">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">XX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> )</span><br/>

<b>thus </b><a NAME="E1:63_3_5_1"/>
(  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E3:63_3_5"><i><font color="Green" title="E40">A29</font></i></a></i>; <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>
<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">Y</font>;<br/>
<a NAME="E6:63_3_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">XX</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E4:63"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E1:63_3_5"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:63_3_5"/><i><font color="Green" title="E42">A31</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z1</font> &amp; <font color="Maroon" title="c9">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:63_3_5"/>
<font color="Maroon" title="c9">Z1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_3.html#E1:63_3_5"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="card_3.html#E8:63_3_5"><i><font color="Green" title="E42">A31</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:63_3_5"/><b>then </b>
<font color="Maroon" title="c7">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_3.html#E3:63_3_5"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="card_3.html#E8:63_3_5"><i><font color="Green" title="E42">A31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:63_3_5"/><i><font color="Green" title="E43">A32</font></i>: 
( <font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> &amp; ( for <font color="Olive" title="b1">Z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Maroon" title="c10">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z2</font> ) )
 <b>by </b><i><a class="txt" href="card_3.html#E5:63"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="card_3.html#E4:63_3_5"><i><font color="Green" title="E41">A30</font></i></a></i>;<br/>
<a NAME="E13:63_3_5"/><i><font color="Green" title="E44">A33</font></i>: 
<font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E3:63_3_5"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="card_3.html#E12:63_3_5"><i><font color="Green" title="E43">A32</font></i></a></i>;<br/>
<a NAME="E14:63_3_5"/><b>then </b>
( <font color="Maroon" title="c10">Z1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z1</font> &amp; <font color="Maroon" title="c10">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">Z1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="card_3.html#E4:63"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="card_3.html#E3:63_3_5"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="card_3.html#E12:63_3_5"><i><font color="Green" title="E43">A32</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:63_3_5"/><i><font color="Green" title="E45">A34</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> ) )
 <b>by </b><i><a class="txt" href="card_3.html#E8:63"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="wellord1.html#T9" title="WELLORD1:th.9">WELLORD1:9</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Y</font> )</span><br/>

<b>thus </b><a NAME="E17:63_3_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="card_3.html#E16:63_3_5"><i><font color="Green" title="E45">A34</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Y</font></span><br/>

<b>assume </b><a NAME="E18:63_3_5"/><i><font color="Green" title="E46">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font>;<br/>
<a NAME="E20:63_3_5"/><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E16:63_3_5"><i><font color="Green" title="E45">A34</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:63_3_5"/><i><font color="Green" title="E48">A37</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E18:63_3_5"><i><font color="Green" title="E46">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:63_3_5"/><b>then </b><i><font color="Green" title="E49">A38</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_5_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c13">Z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_3_5_2"/><i><font color="Green" title="E50">A39</font></i>: 
( <font color="Maroon" title="c13">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Z2</font> &amp; not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Z2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E3:63_3_5_2"/>
<font color="Maroon" title="c13">Z2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="card_3.html#E21:63_3_5"><i><font color="Green" title="E48">A37</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_5_2"><i><font color="Green" title="E50">A39</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:63_3_5_2"/><b>then </b>
<font color="Maroon" title="c13">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="txt" href="card_3.html#E3:63_3_5"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_5_2"><i><font color="Green" title="E50">A39</font></i></a></i>;<br/><a NAME="E5:63_3_5_2"/><b>then </b>
<font color="Maroon" title="c10">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Z2</font>
 <b>by </b><i><a class="txt" href="card_3.html#E12:63_3_5"><i><font color="Green" title="E43">A32</font></i></a></i>;<br/><b>hence </b><a NAME="E6:63_3_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_3.html#E20:63_3_5"><i><font color="Green" title="E47">A36</font></i></a>, <a class="txt" href="card_3.html#E2:63_3_5_2"><i><font color="Green" title="E50">A39</font></i></a></i>; <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>
<a NAME="E24:63_3_5"/><b>then </b><i><font color="Green" title="E50">A40</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E13:63_3_5"><i><font color="Green" title="E44">A33</font></i></a>, <a class="txt" href="card_3.html#E20:63_3_5"><i><font color="Green" title="E47">A36</font></i></a>, <a class="txt" href="card_3.html#E22:63_3_5"><i><font color="Green" title="E49">A38</font></i></a></i>;<br/>
<a NAME="E25:63_3_5"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E21:63_3_5"><i><font color="Green" title="E48">A37</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E26:63_3_5"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E16:63_3_5"><i><font color="Green" title="E45">A34</font></i></a>, <a class="txt" href="card_3.html#E22:63_3_5"><i><font color="Green" title="E49">A38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E27:63_3_5"/>
contradiction
 <b>by </b><i><a class="txt" href="card_3.html#E9:63"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="card_3.html#E22:63_3_5"><i><font color="Green" title="E49">A38</font></i></a>, <a class="txt" href="card_3.html#E24:63_3_5"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>; <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>
<a NAME="E14:63"/><b>then </b>
<font color="Maroon" title="c5">Q</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D2" title="WELLORD2:def.2">WELLORD2:def 2</a></i>;<br/>
<a NAME="E15:63"/><b>then </b>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c5">Q</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T50" title="WELLORD1:th.50">WELLORD1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E17:63"/><i><font color="Green" title="E39">A41</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E18:63"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E19:63"/><b>then </b><i><font color="Green" title="E40">A42</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> ) ) ) )
 
<b>by </b><i><a class="txt" href="card_3.html#E12:63"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_3.html#E17:63"><i><font color="Green" title="E39">A41</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E20:63"/><i><font color="Green" title="E41">A43</font></i>: 
for <font color="Olive" title="b1">Z</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Z</font>, <font color="Maroon" title="c8">x</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="c7">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> implies <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> )</span><br/>



<b>assume </b><a NAME="E1:63_4"/><i><font color="Green" title="E42">A44</font></i>: 
( <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font></span><br/>

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> )</span><br/>

<b>assume </b><a NAME="E2:63_4"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font></span><br/>

<a NAME="E3:63_4"/><b>then </b><i><font color="Green" title="E43">A45</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_4_1"/><i><font color="Green" title="E44">A46</font></i>: 
( <font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font></span><br/><a NAME="E3:63_4_1"/>
( <font color="Maroon" title="c10">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> or <font color="Maroon" title="c7">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z1</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:63"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="card_3.html#E1:63_4"><i><font color="Green" title="E42">A44</font></i></a>, <a class="txt" href="card_3.html#E2:63_4_1"><i><font color="Green" title="E44">A46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:63_4_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="txt" href="card_3.html#E1:63_4"><i><font color="Green" title="E42">A44</font></i></a>, <a class="txt" href="card_3.html#E2:63_4_1"><i><font color="Green" title="E44">A46</font></i></a></i>; <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>
<b>hence </b><a NAME="E5:63_4"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_3.html#E1:63_4"><i><font color="Green" title="E42">A44</font></i></a>, <a class="txt" href="card_3.html#E3:63_4"><i><font color="Green" title="E43">A45</font></i></a></i>; <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>
<a NAME="E21:63"/><i><font color="Green" title="E42">A47</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:63_5"/><i><font color="Green" title="E43">A48</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:63_5"/>
<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> )</span><br/>

<a NAME="E1:63_5_1"/>
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="card_3.html#E1:63_5"><i><font color="Green" title="E43">A48</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:63_5_1"/><i><font color="Green" title="E44">A49</font></i>: 
( <font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><br/>

<b>thus </b><a NAME="E3:63_5_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">f</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:63_5_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:63_5_1_1"/><i><font color="Green" title="E45">A50</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c10">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_3.html#E2:63_5_1"><i><font color="Green" title="E44">A49</font></i></a>, <a class="txt" href="card_3.html#E3:63_5_1_1"><i><font color="Green" title="E45">A50</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:63_5_1_1"/><i><font color="Green" title="E46">A51</font></i>: 
( <font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c11">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E1:63_5"><i><font color="Green" title="E43">A48</font></i></a>, <a class="txt" href="card_3.html#E2:63_5_1"><i><font color="Green" title="E44">A49</font></i></a>, <a class="txt" href="card_3.html#E3:63_5_1_1"><i><font color="Green" title="E45">A50</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E6:63_5_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c7">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">B</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">B</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E1:63_5"><i><font color="Green" title="E43">A48</font></i></a>, <a class="txt" href="card_3.html#E2:63_5_1"><i><font color="Green" title="E44">A49</font></i></a>, <a class="txt" href="card_3.html#E3:63_5_1_1"><i><font color="Green" title="E45">A50</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E7:63_5_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="card_3.html#E1:63_5"><i><font color="Green" title="E43">A48</font></i></a>, <a class="txt" href="card_3.html#E5:63_5_1_1"><i><font color="Green" title="E46">A51</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:63_5_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>; <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>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<b>assume </b><a NAME="E4:63_5_1"/><i><font color="Green" title="E45">A52</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">f</font></span><br/>

<a NAME="E5:63_5_1"/><b>then </b><i><font color="Green" title="E46">A53</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:63_5_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E11:63"><i><font color="Green" title="E37">A8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:63_5_1"/><i><font color="Green" title="E47">A54</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c10">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="card_3.html#E8:63_5_1"><i><font color="Green" title="E47">A54</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E10:63_5_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c7">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="card_3.html#E1:63_5"><i><font color="Green" title="E43">A48</font></i></a>, <a class="txt" href="card_3.html#E5:63_5_1"><i><font color="Green" title="E46">A53</font></i></a>, <a class="txt" href="card_3.html#E8:63_5_1"><i><font color="Green" title="E47">A54</font></i></a></i>;<br/>
<a NAME="E11:63_5_1"/><b>then </b>
( <font color="Maroon" title="c11">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c11">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="card_3.html#E1:63_5"><i><font color="Green" title="E43">A48</font></i></a>, <a class="txt" href="card_3.html#E4:63_5_1"><i><font color="Green" title="E45">A52</font></i></a>, <a class="txt" href="card_3.html#E8:63_5_1"><i><font color="Green" title="E47">A54</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E12:63_5_1"/><b>then </b>
<font color="Maroon" title="c11">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E13:63_5_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="card_3.html#E8:63_5_1"><i><font color="Green" title="E47">A54</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>; <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>
<b>hence </b><a NAME="E3:63_5"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>; <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>
<a NAME="E22:63"/><i><font color="Green" title="E43">A55</font></i>: 
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<b>assume </b><a NAME="E1:63_6"/><i><font color="Green" title="E44">A56</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:63_6"/>
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="card_3.html#E1:63_6"><i><font color="Green" title="E44">A56</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:63_6"/><i><font color="Green" title="E45">A57</font></i>: 
( <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">XX</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E4:63"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:63_6"/>
<font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E20:63"><i><font color="Green" title="E41">A43</font></i></a>, <a class="txt" href="card_3.html#E5:63_6"><i><font color="Green" title="E45">A57</font></i></a></i>;<br/>
<a NAME="E7:63_6"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">Z</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E1:63"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="card_3.html#E4:63"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="card_3.html#E5:63_6"><i><font color="Green" title="E45">A57</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E8:63_6"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="txt" href="card_3.html#E21:63"><i><font color="Green" title="E42">A47</font></i></a>, <a class="txt" href="card_3.html#E1:63_6"><i><font color="Green" title="E44">A56</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E9:63_6"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="card_3.html#T60" target="_self" title="CARD_3:th.60">Th60</a></i>; <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>
<a NAME="E23:63"/>
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font>, <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_3.html#E19:63"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E24:63"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E25:63"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_3.html#E22:63"><i><font color="Green" title="E43">A55</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
