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

<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font>, <font color="Maroon" title="c3">Y1</font>, <font color="Maroon" title="c4">Y2</font> be   <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c2">X2</font> <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c4">Y2</font> <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a> <font color="Maroon" title="c3">Y1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
( <font color="Maroon" title="c2">X2</font> <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c4">Y2</font> <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a> <font color="Maroon" title="c3">Y1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> </span><br/>

<a NAME="E2:57"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c2">X2</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <font color="Maroon" title="c4">Y2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">l</font>, <font color="Maroon" title="c6">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="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> or  not <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> or  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> &amp; <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y2</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="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">l1</font>, <font color="Maroon" title="c8">l2</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:57_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
( <font color="Maroon" title="c7">l1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c8">l2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c7">l1</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c8">l2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">r2</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E5:57_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
( <font color="Maroon" title="c9">r1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c9">r1</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<a NAME="E6:57_1"/>
( <font color="Maroon" title="c1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="surreal0.html#T45" onmouseover="rs('surreal0/T45')" onmouseout="rh()">SURREAL0:45</a></span>;<br/>
<a NAME="E7:57_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:57_1"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E52">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>
<span class="kw">consider </span><font color="Maroon" title="c5">M</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E4:57"/><span class="lab"><font color="Green" title="E50">A6</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="c2">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">Y2</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="c5">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="E5:57"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c4">Y2</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="c5">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E49">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">hence </span><a NAME="E6:57"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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>
