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

<span class="kw">let </span><font color="Maroon" title="c1">x</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">nX</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">Y</font> being   <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; ( ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b2">nX</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="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> ) or ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b2">nX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</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> ) ) holds <br/> <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Olive" title="b1">X</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>,<font color="Olive" title="b3">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Olive" title="b3">Y</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>,<font color="Olive" title="b2">nX</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">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">set </span><font color="Maroon" title="c3">Inv</font> =  <a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</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>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">X1</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">  for <font color="Olive" title="b1">Y</font> being   <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; ( ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">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="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> ) or ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</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> ) ) holds <br/> <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Olive" title="b1">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Olive" title="b1">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">Y</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="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; ( ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">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="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> ) or ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</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> ) ) implies  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">and </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
( ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">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="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> ) or ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</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> ) )
 ; <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="c4">X</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>,<font color="Maroon" title="c6">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)</span><br/>

<span class="kw">thus </span><a NAME="E3:54"/>
 <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI: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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>) or <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">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"> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:54_1"/><span class="lab"><font color="Green" title="E47">A3</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="c8">x1</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )
 <span class="kw">and </span><br/><a NAME="E4:54_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c7">o</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="c8">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <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>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">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="c8">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="E5:54_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon" title="c8">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a></span>;<br/>
<a NAME="E6:54_1"/>
 not <font color="Maroon" title="c8">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a></span>;<br/>
<a NAME="E7:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c8">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="ref" href="surreali.html#D14" target="_self" onmouseover="rs('surreali/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E8:54_1"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c8">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a></span>;<br/>
<a NAME="E9:54_1"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
;<br/>
<a NAME="E10:54_1"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E11:54_1"/>
( <font color="Maroon" title="c8">x1</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"><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> or <font color="Maroon" title="c8">x1</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"><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="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E12:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E13:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <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>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<a NAME="E14:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <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>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c9">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E15:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c9">y1</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E9:54_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:54_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</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">let </span><font color="Maroon" title="c7">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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI: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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>) or <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E4:54"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c6">Y</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:54"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">y1</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>,<font color="Maroon" title="c5">X1</font>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c8">y1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:54"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c7">o</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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <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>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c9">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/>
<a NAME="E11:54"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
( <font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c11">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E12:54"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
 not <font color="Maroon" title="c11">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E49">A12</font></span></a></span>;<br/>
<a NAME="E13:54"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c11">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="surreali.html#D14" target="_self" onmouseover="rs('surreali/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E14:54"/>
<font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E48">A11</font></span></a></span>;<br/>
<a NAME="E15:54"/><span class="kw">then </span>
<font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <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>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<a NAME="E16:54"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</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="c11">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <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>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">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="c11">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="#E9:54"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E13:54"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:54"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<font color="Maroon" title="c4">X</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>,<font color="Maroon" title="c6">Y</font>)
 <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>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E47">A10</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>
