<?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">  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">OA</font> =  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E1:137"/><span class="lab"><font color="Green" title="E106">A1</font></span>: 
<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#K9" title="SURREALN:func.9">No_uOrdinal_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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E2:137"/><span class="lab"><font color="Green" title="E107">A2</font></span>: 
 <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="c1">A</font></span>)</span> <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="ref" href="surrealn.html#T73" target="_self" onmouseover="rs('surrealn/T73')" onmouseout="rh()">Th73</a></span>;<br/>
<a NAME="E3:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A3</font></span>: 
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</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="c1">A</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/>
<a NAME="E4:137"/>
 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> <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#K9" title="SURREALN:func.9">No_uOrdinal_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="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>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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="137_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">o</font> <a href="hidden.html#R2" title="HIDDEN: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#K9" title="SURREALN:func.9">No_uOrdinal_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="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>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">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="E1:137_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN: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#K9" title="SURREALN:func.9">No_uOrdinal_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="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>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">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/>

<a NAME="E2:137_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><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="E3:137_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>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">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="#E3:137"><span class="lab"><font color="Green" title="E108">A3</font></span></a>, <a class="ref" href="ordinal1.html#T6" onmouseover="rs('ordinal1/T6')" onmouseout="rh()">ORDINAL1:6</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="E5:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_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="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> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137"><span class="lab"><font color="Green" title="E106">A1</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="c4">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_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="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="E7:137"/><span class="lab"><font color="Green" title="E110">A5</font></span>: 
 <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T65" target="_self" onmouseover="rs('surrealn/T65')" onmouseout="rh()">Th65</a></span>;<br/>
<a NAME="E8:137"/>
 <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="ref" href="surrealn.html#T73" target="_self" onmouseover="rs('surrealn/T73')" onmouseout="rh()">Th73</a></span>;<br/>
<a NAME="E9:137"/><span class="kw">then </span>
<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#K9" title="SURREALN:func.9">No_uOrdinal_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="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T32" onmouseover="rs('surrealo/T32')" onmouseout="rh()">SURREALO:32</a></span>;<br/>
<a NAME="E10:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A6</font></span>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:137"><span class="lab"><font color="Green" title="E110">A5</font></span></a>, <a class="ref" href="surrealo.html#T29" onmouseover="rs('surrealo/T29')" onmouseout="rh()">SURREALO:29</a></span>;<br/>
<a NAME="E11:137"/><span class="lab"><font color="Green" title="E112">A7</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c4">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:137"><span class="lab"><font color="Green" title="E109">A4</font></span></a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E12:137"/>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A8</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:137"><span class="lab"><font color="Green" title="E107">A2</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="ordinal1.html#T21" onmouseover="rs('ordinal1/T21')" onmouseout="rh()">ORDINAL1:21</a></span>;<br/>
<a NAME="E14:137"/>
 for <font color="Olive" title="b1">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> holds <br/> <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="137_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</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="c5">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> implies  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:137_2"/><span class="lab"><font color="Green" title="E114">A9</font></span>: 
( <font color="Maroon" title="c5">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> &amp;  not  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</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="E2:137_2"/>
(  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><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>, <a class="txt" href="#E1:137_2"><span class="lab"><font color="Green" title="E114">A9</font></span></a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a></span>;<br/>
<a NAME="E3:137_2"/><span class="kw">then </span>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<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/>
<a NAME="E4:137_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A10</font></span>: 
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:137"><span class="lab"><font color="Green" title="E111">A6</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E5:137_2"/>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137_2"><span class="lab"><font color="Green" title="E114">A9</font></span></a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="ref" href="ordinal1.html#T22" onmouseover="rs('ordinal1/T22')" onmouseout="rh()">ORDINAL1:22</a></span>;<br/>
<a NAME="E6:137_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">y</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="c5">y</font></span>)</span> &amp;  <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="c5">y</font></span>)</span> <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="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E7:137_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T76" target="_self" onmouseover="rs('surrealn/T76')" onmouseout="rh()">Th76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:137_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137_2"><span class="lab"><font color="Green" title="E114">A9</font></span></a>, <a class="txt" href="#E4:137_2"><span class="lab"><font color="Green" title="E115">A10</font></span></a>, <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="E15:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A11</font></span>: 
 <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:137"><span class="lab"><font color="Green" title="E113">A8</font></span></a>, <a class="ref" href="surrealo.html#D5" onmouseover="rs('surrealo/D5')" onmouseout="rh()">SURREALO:def 5</a>, <a class="txt" href="#E11:137"><span class="lab"><font color="Green" title="E112">A7</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E16:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A12</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealo.html#K3" title="SURREALO:func.3">born_eq_set</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:137"><span class="lab"><font color="Green" title="E109">A4</font></span></a>, <a class="ref" href="surrealo.html#D6" onmouseover="rs('surrealo/D6')" onmouseout="rh()">SURREALO:def 6</a></span>;<br/>
<a NAME="E17:137"/><span class="lab"><font color="Green" title="E116">A13</font></span>: 
<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span> is  <a href="surrealo.html#V3" title="SURREALO:attr.3">uniq-surreal-membered</a> 
 
;<br/>
<a NAME="E18:137"/>
 for <font color="Olive" title="b1">z</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealo.html#K3" title="SURREALO:func.3">born_eq_set</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Olive" title="b1">z</font></span>)</span> is  <a href="surrealo.html#V3" title="SURREALO:attr.3">uniq-surreal-membered</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">z</font> holds <br/><span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Olive" title="b1">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="137_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</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="c5">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealo.html#K3" title="SURREALO:func.3">born_eq_set</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span> is  <a href="surrealo.html#V3" title="SURREALO:attr.3">uniq-surreal-membered</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> implies <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:137_3"/><span class="lab"><font color="Green" title="E117">A14</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealo.html#K3" title="SURREALO:func.3">born_eq_set</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span> is  <a href="surrealo.html#V3" title="SURREALO:attr.3">uniq-surreal-membered</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> )
 <span class="kw">and </span><br/><a NAME="E2:137_3"/><span class="lab"><font color="Green" title="E118">A15</font></span>: 
 not <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</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"> contradiction</span><br/>

<a NAME="E3:137_3"/><span class="lab"><font color="Green" title="E119">A16</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">z</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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:137"><span class="lab"><font color="Green" title="E114">A11</font></span></a>, <a class="txt" href="#E1:137_3"><span class="lab"><font color="Green" title="E117">A14</font></span></a>, <a class="ref" href="surrealo.html#D6" onmouseover="rs('surrealo/D6')" onmouseout="rh()">SURREALO:def 6</a></span>;<br/>
<a NAME="E4:137_3"/><span class="kw">then </span>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</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;  <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="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T43" onmouseover="rs('surreal0/T43')" onmouseout="rh()">SURREAL0:43</a></span>;<br/>
<a NAME="E5:137_3"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">xR</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>( <font color="Olive" title="b1">xR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> &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#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b1">xR</font> &amp; <font color="Olive" title="b1">xR</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> ) or  ex <font color="Olive" title="b1">yL</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>( <font color="Olive" title="b1">yL</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="c5">z</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#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Olive" title="b1">yL</font> &amp; <font color="Olive" title="b1">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c5">z</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T13" onmouseover="rs('surrealo/T13')" onmouseout="rh()">SURREALO:13</a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">zL</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E7:137_3"/><span class="lab"><font color="Green" title="E120">A17</font></span>: 
( <font color="Maroon" title="c6">zL</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="c5">z</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#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c6">zL</font> &amp; <font color="Maroon" title="c6">zL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c5">z</font> )
 <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="E8:137_3"/>
<font color="Maroon" title="c6">zL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:137_3"><span class="lab"><font color="Green" title="E120">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:137_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A18</font></span>: 
<font color="Maroon" title="c6">zL</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137_3"><span class="lab"><font color="Green" title="E117">A14</font></span></a></span>;<br/>
<a NAME="E10:137_3"/>
<font color="Maroon" title="c6">zL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:137_3"><span class="lab"><font color="Green" title="E120">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:137_3"/><span class="kw">then </span>
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c6">zL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">z</font> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">z</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="txt" href="#E3:137_3"><span class="lab"><font color="Green" title="E119">A16</font></span></a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E12:137_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">zL</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="c6">zL</font></span>)</span> &amp;  <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="c6">zL</font></span>)</span> <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="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a>, <a class="ref" href="ordinal1.html#T22" onmouseover="rs('ordinal1/T22')" onmouseout="rh()">ORDINAL1:22</a></span>;<br/>
<a NAME="E13:137_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">zL</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="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T76" target="_self" onmouseover="rs('surrealn/T76')" onmouseout="rh()">Th76</a>, <a class="txt" href="#E7:137_3"><span class="lab"><font color="Green" title="E120">A17</font></span></a></span>;<br/>
<a NAME="E14:137_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A19</font></span>: 
<font color="Maroon" title="c6">zL</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="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:137_3"><span class="lab"><font color="Green" title="E121">A18</font></span></a>, <a class="ref" href="surrealo.html#T50" onmouseover="rs('surrealo/T50')" onmouseout="rh()">SURREALO:50</a></span>;<br/>
<a NAME="E15:137_3"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/>
<a NAME="E16:137_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal7.html#T69" onmouseover="rs('ordinal7/T69')" onmouseout="rh()">ORDINAL7:69</a></span>;<br/>
<a NAME="E17:137_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="ordinal7.html#K5" title="ORDINAL7:func.5">(+)</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="txt" href="#E2:137_3"><span class="lab"><font color="Green" title="E118">A15</font></span></a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="txt" href="#E7:137_3"><span class="lab"><font color="Green" title="E120">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">w</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E19:137_3"/><span class="lab"><font color="Green" title="E123">A20</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T47" onmouseover="rs('surrealo/T47')" onmouseout="rh()">SURREALO:47</a></span>;<br/>
<span class="kw">thus </span><a NAME="E20:137_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:137_3"><span class="lab"><font color="Green" title="E123">A20</font></span></a>, <a class="txt" href="#E7:137_3"><span class="lab"><font color="Green" title="E120">A17</font></span></a>, <a class="txt" href="#E1:137_3"><span class="lab"><font color="Green" title="E117">A14</font></span></a>, <a class="txt" href="#E14:137_3"><span class="lab"><font color="Green" title="E122">A19</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="E19:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A21</font></span>: 
<font color="Maroon" title="c4">x</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:137"><span class="lab"><font color="Green" title="E115">A12</font></span></a>, <a class="txt" href="#E17:137"><span class="lab"><font color="Green" title="E116">A13</font></span></a>, <a class="ref" href="surrealo.html#T49" onmouseover="rs('surrealo/T49')" onmouseout="rh()">SURREALO:49</a></span>;<br/>
<a NAME="E20:137"/>
 <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<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="E21:137"/><span class="kw">then </span>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> <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="txt" href="#E10:137"><span class="lab"><font color="Green" title="E111">A6</font></span></a>, <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="E22:137"/>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:137"><span class="lab"><font color="Green" title="E117">A21</font></span></a>, <a class="ref" href="surrealo.html#T50" onmouseover="rs('surrealo/T50')" onmouseout="rh()">SURREALO:50</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>
