<?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">Inv</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">Inv</font> is <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></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> <font color="Maroon" title="c1">x</font></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> <a href="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</a>  holds <br/>(  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Inv</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">Inv</font> is <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></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> <font color="Maroon" title="c1">x</font></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> <a href="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</a>  implies (  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c2">Inv</font> is <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></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> <font color="Maroon" title="c1">x</font></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> <a href="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:17"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> or <font color="Maroon" title="c3">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</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"> <font color="Maroon" title="c3">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/>
<a NAME="E4:17_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D5" target="_self" onmouseover="rs('surreali/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>;<br/>
<a NAME="E6:17_1"/>
<span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T9" target="_self" onmouseover="rs('surreali/T9')" onmouseout="rh()">Th9</a>, <a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17_1"/>
<font color="Maroon" title="c3">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E16">A2</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><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> or <font color="Maroon" title="c3">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:17"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</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"> <font color="Maroon" title="c3">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:17"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/>
<a NAME="E6:17"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D6" target="_self" onmouseover="rs('surreali/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/>
<a NAME="E8:17"/>
<span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">Inv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T9" target="_self" onmouseover="rs('surreali/T9')" onmouseout="rh()">Th9</a>, <a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:17"/>
<font color="Maroon" title="c3">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E16">A3</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>
