<?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">  for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> holds <br/> <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b2">Y</font>) <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Olive" title="b2">Y</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be    <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="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> implies  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">Y</font>) <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:55"/><span class="lab"><font color="Green" title="E46">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="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">Y</font>) <a href="surrealo.html#R3" title="SURREALO:pred.3">&lt;=_</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">z</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="surrealo.html#D3" onmouseover="rs('surrealo/D3')" onmouseout="rh()">SURREALO:def 3</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">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">Y</font>) or  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) &amp; <font color="Olive">b<sub>1</sub></font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:55"/>
<font color="Maroon" title="c5">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">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">  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) &amp; <font color="Olive">b<sub>1</sub></font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E4:55"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <span class="kw">and </span><br/><a NAME="E5:55"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D15" target="_self" onmouseover="rs('surreali/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<a NAME="E6:55"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="surreali.html#D15" target="_self" onmouseover="rs('surreali/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<a NAME="E7:55"/>
 <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="surrealr.html#T10" onmouseover="rs('surrealr/T10')" onmouseout="rh()">SURREALR:10</a></span>;<br/>
<a NAME="E8:55"/><span class="kw">then </span>
<font color="Maroon" title="c6">x1</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> <font color="Maroon" title="c1">x</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c6">x1</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> <font color="Maroon" title="c2">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/>
<a NAME="E9:55"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/>
<a NAME="E10:55"/><span class="kw">then </span>
<a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/>
<a NAME="E11:55"/><span class="kw">then </span>
<font color="Maroon" title="c5">z</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:55"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">Y</font>) &amp; <font color="Olive">b<sub>1</sub></font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E49">A4</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>
