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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is  <a href="surrealn.html#V3" title="SURREALN:attr.3">No_ordinal</a>  implies  ex <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Olive" title="b1">A</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span><font color="Maroon" title="c1">x</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> $1;<br/>
<a NAME="E1:127"/>
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E2:127"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T69" target="_self" onmouseover="rs('surrealn/T69')" onmouseout="rh()">Th69</a></span>;<br/>
<a NAME="E3:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1</font></span>: 
 ex <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">Min</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E5:127"/><span class="lab"><font color="Green" title="E102">A2</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">Min</font>] &amp; (  for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/><font color="Maroon" title="c2">Min</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" onmouseover="rs('ordinal1/S1')" onmouseout="rh()">ORDINAL1:sch 1</a>(<a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E101">A1</font></span></a>);<br/></span>
<span class="kw">assume </span><a NAME="E6:127"/><span class="lab"><font color="Green" title="E103">A3</font></span>: 
<font color="Maroon" title="c1">x</font> is  <a href="surrealn.html#V3" title="SURREALN:attr.3">No_ordinal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Olive" title="b1">A</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">Min</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font>;<br/>
<a NAME="E7:127"/>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font></span>)</span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D20" onmouseover="rs('surreal0/D20')" onmouseout="rh()">SURREAL0:def 20</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font></span>)</span> or  not <font color="Maroon" title="c5">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:127_1"/><span class="lab"><font color="Green" title="E104">A4</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c4">a</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c5">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:127_1"/><span class="lab"><font color="Green" title="E105">A5</font></span>: 
( <font color="Maroon" title="c6">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Min</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c6">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E104">A4</font></span></a>, <a class="ref" href="surrealn.html#T71" target="_self" onmouseover="rs('surrealn/T71')" onmouseout="rh()">Th71</a></span>;<br/>
<a NAME="E4:127_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1"><span class="lab"><font color="Green" title="E105">A5</font></span></a>, <a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E104">A4</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:127_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127"><span class="lab"><font color="Green" title="E102">A2</font></span></a>, <a class="txt" href="#E3:127_1"><span class="lab"><font color="Green" title="E105">A5</font></span></a>, <a class="ref" href="ordinal1.html#T5" onmouseover="rs('ordinal1/T5')" onmouseout="rh()">ORDINAL1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:127"/><span class="kw">then </span>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font></span>)</span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E103">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:127"/>
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c2">Min</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127"><span class="lab"><font color="Green" title="E102">A2</font></span></a>, <a class="ref" href="surreal0.html#T43" onmouseover="rs('surreal0/T43')" onmouseout="rh()">SURREAL0:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
