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<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> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c2">y</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">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">Nx</font> = <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>;<br/>
<span class="kw">assume </span><a NAME="E1:59"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</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="c2">y</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">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> )</span><br/>

<a NAME="E2:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T18" target="_self" onmouseover="rs('surreali/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E3:59"/><span class="kw">then </span>
 <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>) <a href="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T35" target="_self" onmouseover="rs('surreali/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E4:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</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"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T31" onmouseover="rs('surrealo/T31')" onmouseout="rh()">SURREALO:31</a></span>;<br/>
<a NAME="E5:59"/>
 <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>) <a href="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="surreali.html#T35" target="_self" onmouseover="rs('surreali/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E6:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</a> <span class="p1">(<span class="default"><span class="p2"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="surrealo.html#T31" onmouseover="rs('surrealo/T31')" onmouseout="rh()">SURREALO:31</a></span>;<br/>
<a NAME="E7:59"/>
(  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>) <a href="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>) &amp;  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>) <a href="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="surreali.html#T35" target="_self" onmouseover="rs('surreali/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E8:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="surrealo.html#R4" title="SURREALO:pred.4">&lt;==&gt;</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T31" onmouseover="rs('surrealo/T31')" onmouseout="rh()">SURREALO:31</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:59"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<font color="Maroon" title="c2">y</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">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.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"> <font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font></span><br/>

<a NAME="E10:59"/>
<font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <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">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="surreali.html#T36" target="_self" onmouseover="rs('surreali/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:59"/>
<font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T29" onmouseover="rs('surrealo/T29')" onmouseout="rh()">SURREALO:29</a>, <a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E52">A5</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>
