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

<span class="kw">set </span><font color="Maroon" title="c2">L</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> ;<br/>
<a NAME="E1:100_1_1"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  or <font color="Maroon" title="c4">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_1_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </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"> <font color="Maroon" title="c4">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> </span><br/>

<a NAME="E2:100_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:100_1_1_1"/>
<font color="Maroon" title="c4">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:100_1_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_2"><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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  or <font color="Maroon" title="c4">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_1_1_2"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </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"> <font color="Maroon" title="c4">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> </span><br/>

<a NAME="E2:100_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:100_1_1_2"/>
<font color="Maroon" title="c4">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">L</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> , <font color="Maroon" title="c5">R</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  as   <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1"><span class="lab"><font color="Green" title="E81">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">M</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E5:100_1_1"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
 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="c4">L</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">R</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">M</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">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T47" onmouseover="rs('surreal0/T47')" onmouseout="rh()">SURREAL0:47</a></span>;<br/>
<a NAME="E6:100_1_1"/>
<font color="Maroon" title="c4">L</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <font color="Maroon" title="c5">R</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">l</font>, <font color="Maroon" title="c8">r</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="c7">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">L</font> or  not <font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">R</font> or  not <font color="Maroon" title="c8">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c7">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_1_1_3"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
( <font color="Maroon" title="c7">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">R</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">  not <font color="Maroon" title="c8">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c7">l</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">nL</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:100_1_1_3"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_3"><span class="lab"><font color="Green" title="E83">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">nR</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:100_1_1_3"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">nR</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_3"><span class="lab"><font color="Green" title="E83">A3</font></span></a></span>;<br/>
<a NAME="E6:100_1_1_3"/>
(  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp;  <a href="surrealr.html#K2" title="SURREALR:func.2">-</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T10" onmouseover="rs('surrealr/T10')" onmouseout="rh()">SURREALR:10</a>, <a class="ref" href="surreali.html#D8" onmouseover="rs('surreali/D8')" onmouseout="rh()">SURREALI:def 8</a></span>;<br/>
<a NAME="E7:100_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A6</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/>
<a NAME="E8:100_1_1_3"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">nR</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a>, <a class="ref" href="surreali.html#D8" onmouseover="rs('surreali/D8')" onmouseout="rh()">SURREALI:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:100_1_1_3"/>
 not <font color="Maroon" title="c8">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c7">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100_1_1_3"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E5:100_1_1_3"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E7:100_1_1_3"><span class="lab"><font color="Green" title="E86">A6</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="E7:100_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">L</font>,<font color="Maroon" title="c5">R</font></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="c6">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:100_1_1"><span class="lab"><font color="Green" title="E82">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="c7">LR</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">L</font>,<font color="Maroon" title="c5">R</font></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/>
<span class="kw">take </span>
<font color="Maroon" title="c7">LR</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#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c7">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c7">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:100_1_1"/>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c7">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span>  &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c7">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </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"> verum</span><br/>


</div>
