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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">M</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Olive" title="b1">M</font> holds <br/>(  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c2">M</font> implies (  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">A</font> =  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<a NAME="E3:35"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">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="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="card_5.html#T7" target="_self" title="CARD_5:th.7">Th16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">N</font> being   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a><span class="kw"> such that </span><br/><a NAME="E5:35"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c4">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E7:35"/>
 for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a> <font color="Olive" title="b1">C</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T9" target="_self" title="CARD_5:th.9">Th18</a></span>;<br/>
<a NAME="E8:35"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="card_5.html#T6" target="_self" title="CARD_5:th.6">Th15</a></span>;<br/>
<a NAME="E9:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a> <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="ordinal4.html#T37" title="ORDINAL4:th.37">ORDINAL4:37</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E33">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1"/><span class="kw">then </span>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c2">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="card_5.html#D1" target="_self" title="CARD_5:def.1">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="card_1.html#T4" title="CARD_1:th.4">CARD_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:35"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E27">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">A</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></span>;<br/>
<a NAME="E13:35"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T18" title="ORDINAL2:th.18">ORDINAL2:18</a></span>;<br/>
<a NAME="E14:35"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="ordinal2.html#T22" title="ORDINAL2:th.22">ORDINAL2:22</a></span>;<br/>
<a NAME="E15:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:35"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E17:35"/>
<font color="Maroon" title="c5">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_2"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:35_2"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">Y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E37">A11</font></span></a></span>;<br/>
<a NAME="E6:35_2"/>
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E7:35_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:35_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2"><span class="lab"><font color="Green" title="E36">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:35"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
