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

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">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="c1">x</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="c2">y</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> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  implies <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/>
( <font color="Maroon" title="c1">x</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="c2">y</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> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> </span><br/>

<a NAME="E2:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A1</font></span>: 
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">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"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></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"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a></span>;<br/>
<a NAME="E3:24"/>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T37" onmouseover="rs('surreal0/T37')" onmouseout="rh()">SURREAL0:37</a></span>;<br/>
<a NAME="E4:24"/><span class="kw">then </span>
 for <font color="Olive" title="b1">z</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">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="Olive" title="b1">z</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"><font color="Olive" title="b1">z</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> <font color="Maroon" title="c1">x</font> holds <br/> <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <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">z</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:24"/>
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T16" onmouseover="rs('surrealo/T16')" onmouseout="rh()">SURREALO:16</a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">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/>


</div>
