<?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="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> implies  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:103"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">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">  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></span><br/>

<a NAME="E2:103"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">l</font>, <font color="Maroon" title="c4">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="c3">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span> or  not <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c3">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:103_1"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
( <font color="Maroon" title="c3">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp; <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c4">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c3">l</font></span><br/>

<a NAME="E2:103_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">l</font> <a href="tarski.html#R2" title="TARSKI: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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D8" target="_self" onmouseover="rs('surrealn/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E4:103_1"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c3">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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 ;<br/>
<a NAME="E5:103_1"/>
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E6:103_1"/><span class="kw">then </span>
( <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">nL</font></span>)</span> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">nL</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" onmouseover="rs('xreal_1/T76')" onmouseout="rh()">XREAL_1:76</a>, <a class="ref" href="surrealn.html#T58" target="_self" onmouseover="rs('surrealn/T58')" onmouseout="rh()">Th58</a>, <a class="ref" href="surrealn.html#T51" target="_self" onmouseover="rs('surrealn/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E7:103_1"/><span class="kw">then </span>
( <span class="p1">(<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> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T58" target="_self" onmouseover="rs('surrealn/T58')" onmouseout="rh()">Th58</a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E8:103_1"/><span class="kw">then </span>
<span class="p1">(<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> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a> 
 
<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="E9:103_1"/><span class="kw">then </span>
 <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="c5">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> <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> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></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#T10" onmouseover="rs('surrealr/T10')" onmouseout="rh()">SURREALR:10</a></span>;<br/>
<a NAME="E10:103_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A5</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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</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> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <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="#E1:103"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/>
<a NAME="E11:103_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E12:103_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</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> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></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> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T59" target="_self" onmouseover="rs('surrealn/T59')" onmouseout="rh()">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:103_1"/>
 not <font color="Maroon" title="c4">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103_1"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E10:103_1"><span class="lab"><font color="Green" title="E88">A5</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="E3:103"/>
<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#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">l</font>, <font color="Maroon" title="c4">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="c3">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">r</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#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></span>)</span> or  not <font color="Maroon" title="c4">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c3">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:103_2"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
( <font color="Maroon" title="c3">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">r</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#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font></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">  not <font color="Maroon" title="c4">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c3">l</font></span><br/>

<a NAME="E2:103_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</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 class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D8" target="_self" onmouseover="rs('surrealn/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E4:103_2"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 ;<br/>
<a NAME="E5:103_2"/>
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E6:103_2"/><span class="kw">then </span>
( <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">nL</font></span>)</span> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">nL</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" onmouseover="rs('xreal_1/T76')" onmouseout="rh()">XREAL_1:76</a>, <a class="ref" href="surrealn.html#T58" target="_self" onmouseover="rs('surrealn/T58')" onmouseout="rh()">Th58</a>, <a class="ref" href="surrealn.html#T51" target="_self" onmouseover="rs('surrealn/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E7:103_2"/><span class="kw">then </span>
( <span class="p1">(<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> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T58" target="_self" onmouseover="rs('surrealn/T58')" onmouseout="rh()">Th58</a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E8:103_2"/><span class="kw">then </span>
<span class="p1">(<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> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<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="c5">nL</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a> 
 
<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="E9:103_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A8</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"><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> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">nL</font></span>)</span></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</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="c5">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="#E1:103"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/>
<a NAME="E10:103_2"/>
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_2"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:103_2"/>
 not <font color="Maroon" title="c4">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103_2"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="txt" href="#E9:103_2"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="ref" href="surrealn.html#T59" target="_self" onmouseover="rs('surrealn/T59')" onmouseout="rh()">Th59</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>
<span class="kw">hence </span><a NAME="E4:103"/>
 <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="ref" href="surreal0.html#T43" onmouseover="rs('surreal0/T43')" onmouseout="rh()">SURREAL0:43</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>
