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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">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">x</font> being   <a href="surrealn.html#V3" title="SURREALN:attr.3">No_ordinal</a>  <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Olive" title="b1">x</font> &amp; (  for <font color="Olive" title="b2">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b1">x</font> iff  ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b3">B</font> ) ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b1">B</font> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:138"/><span class="lab"><font color="Green" title="E106">A1</font></span>: 
 for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" onmouseover="rs('xboole_0/S1')" onmouseout="rh()">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E3:138"/><span class="lab"><font color="Green" title="E107">A2</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E4:138"/>
 for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> holds <br/> ex <font color="Olive" title="b2">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b2">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Olive" title="b2">O</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="138_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> implies  ex <font color="Olive" title="b1">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Olive" title="b1">O</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:138_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Olive" title="b1">O</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:138_1"/><span class="lab"><font color="Green" title="E108">A3</font></span>: 
( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c4">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:138"><span class="lab"><font color="Green" title="E106">A1</font></span></a></span>;<br/>
<a NAME="E4:138_1"/>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T73" target="_self" onmouseover="rs('surrealn/T73')" onmouseout="rh()">Th73</a></span>;<br/>
<a NAME="E5:138_1"/><span class="kw">then </span>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c4">B</font>
 
<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/>
<span class="kw">hence </span><a NAME="E6:138_1"/>
 ex <font color="Olive" title="b1">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Olive" title="b1">O</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_1"><span class="lab"><font color="Green" title="E108">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:138"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">X</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></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="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138"><span class="lab"><font color="Green" title="E107">A2</font></span></a>, <a class="ref" href="surreal0.html#T46" onmouseover="rs('surreal0/T46')" onmouseout="rh()">SURREAL0:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">X</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> ;<br/>
<a NAME="E7:138"/>
 <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="surrealn.html#V3" title="SURREALN:attr.3">No_ordinal</a>  <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D10" target="_self" onmouseover="rs('surrealn/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">x</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"> (  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> &amp; (  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> iff  ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b2">B</font> ) ) ) )</span><br/>

<a NAME="E9:138"/>
 for <font color="Olive" title="b1">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="138_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">O</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">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"> ( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c5">O</font> implies <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:138_2"/><span class="lab"><font color="Green" title="E109">A5</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c5">O</font> &amp;  not <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O</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/>

<a NAME="E2:138_2"/><span class="lab"><font color="Green" title="E110">A6</font></span>: 
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c5">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c5">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T74" target="_self" onmouseover="rs('surrealn/T74')" onmouseout="rh()">Th74</a></span>;<br/>
<a NAME="E3:138_2"/>
 <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c5">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138_2"><span class="lab"><font color="Green" title="E109">A5</font></span></a>, <a class="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a></span>;<br/>
<a NAME="E4:138_2"/><span class="kw">then </span>
(  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c5">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">x</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:138_2"><span class="lab"><font color="Green" title="E110">A6</font></span></a>, <a class="txt" href="#E1:138_2"><span class="lab"><font color="Green" title="E109">A5</font></span></a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="txt" href="#E2:138"><span class="lab"><font color="Green" title="E106">A1</font></span></a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</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="E5:138_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138_2"><span class="lab"><font color="Green" title="E109">A5</font></span></a>, <a class="ref" href="surrealn.html#T76" target="_self" onmouseover="rs('surrealn/T76')" onmouseout="rh()">Th76</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>
<span class="kw">hence </span><a NAME="E10:138"/>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:138"><span class="lab"><font color="Green" title="E108">A4</font></span></a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</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"> (  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> &amp; (  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> iff  ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b2">B</font> ) ) ) )</span><br/>

<a NAME="E11:138"/>
 <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="c1">A</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="c4">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="138_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">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="c5">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="c1">A</font></span>)</span> or  not <font color="Maroon" title="c6">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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c6">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:138_3"/><span class="lab"><font color="Green" title="E109">A7</font></span>: 
( <font color="Maroon" title="c5">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="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c6">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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:138_3"/><span class="lab"><font color="Green" title="E110">A8</font></span>: 
( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">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="c7">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138_3"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="surrealn.html#T71" target="_self" onmouseover="rs('surrealn/T71')" onmouseout="rh()">Th71</a></span>;<br/>
<a NAME="E4:138_3"/><span class="lab"><font color="Green" title="E111">A9</font></span>: 
(  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c7">B</font> &amp;  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_3"><span class="lab"><font color="Green" title="E110">A8</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a>, <a class="ref" href="surrealn.html#T74" target="_self" onmouseover="rs('surrealn/T74')" onmouseout="rh()">Th74</a>, <a class="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a></span>;<br/>
<a NAME="E5:138_3"/>
(  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> <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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_3"><span class="lab"><font color="Green" title="E110">A8</font></span></a>, <a class="txt" href="#E2:138"><span class="lab"><font color="Green" title="E106">A1</font></span></a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a>, <a class="txt" href="#E4:138_3"><span class="lab"><font color="Green" title="E111">A9</font></span></a></span>;<br/>
<a NAME="E6:138_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c7">B</font> &amp;  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c7">B</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_3"><span class="lab"><font color="Green" title="E110">A8</font></span></a>, <a class="ref" href="surrealn.html#T73" target="_self" onmouseover="rs('surrealn/T73')" onmouseout="rh()">Th73</a>, <a class="txt" href="#E1:138_3"><span class="lab"><font color="Green" title="E109">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:138_3"/>
 not <font color="Maroon" title="c6">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:138"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A10</font></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="c1">A</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="c4">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="c1">A</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="c4">x</font> )
 
;<br/>
<a NAME="E13:138"/>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> <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"><span class="p2">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></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="138_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">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="c5">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> or  not <font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c6">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:138_4"/><span class="lab"><font color="Green" title="E110">A11</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:138_4"/><span class="lab"><font color="Green" title="E111">A12</font></span>: 
( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c7">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:138"><span class="lab"><font color="Green" title="E106">A1</font></span></a>, <a class="txt" href="#E1:138_4"><span class="lab"><font color="Green" title="E110">A11</font></span></a></span>;<br/>
<a NAME="E4:138_4"/>
( <font color="Maroon" title="c5">a</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</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="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_4"><span class="lab"><font color="Green" title="E111">A12</font></span></a>, <a class="ref" href="surrealn.html#T75" target="_self" onmouseover="rs('surrealn/T75')" onmouseout="rh()">Th75</a>, <a class="ref" href="surrealn.html#T73" target="_self" onmouseover="rs('surrealn/T73')" onmouseout="rh()">Th73</a></span>;<br/>
<a NAME="E5:138_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:138_4"/>
 not <font color="Maroon" title="c6">b</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138_4"><span class="lab"><font color="Green" title="E110">A11</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:138"/><span class="kw">then </span>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> <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"><span class="p2">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></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"><font color="Maroon" title="c4">x</font></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> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D10" target="_self" onmouseover="rs('surrealn/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E15:138"/><span class="kw">then </span>
(  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</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="c1">A</font> &amp;  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T43" onmouseover="rs('surreal0/T43')" onmouseout="rh()">SURREAL0:43</a>, <a class="txt" href="#E12:138"><span class="lab"><font color="Green" title="E109">A10</font></span></a>, <a class="ref" href="surrealn.html#T73" target="_self" onmouseover="rs('surrealn/T73')" onmouseout="rh()">Th73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:138"/>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> iff  ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b2">B</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> iff  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b1">B</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E17:138"/>
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> implies  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b1">B</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:138"><span class="lab"><font color="Green" title="E106">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Olive" title="b1">B</font> ) implies <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<span class="kw">given </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><a NAME="E19:138"/><span class="lab"><font color="Green" title="E110">A13</font></span>: 
( <font color="Maroon" title="c6">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c6">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"> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E20:138"/>
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c6">B</font> &amp;  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:138"><span class="lab"><font color="Green" title="E110">A13</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a>, <a class="ref" href="surrealn.html#T74" target="_self" onmouseover="rs('surrealn/T74')" onmouseout="rh()">Th74</a>, <a class="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:138"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:138"><span class="lab"><font color="Green" title="E110">A13</font></span></a>, <a class="txt" href="#E2:138"><span class="lab"><font color="Green" title="E106">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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