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

<span class="kw">let </span><font color="Maroon" title="c1">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><br/>  for <font color="Olive" title="b2">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">Inv</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Inv</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Olive" title="b3">Inv</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">Inv</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Inv</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Olive" title="b2">Inv</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</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">Inv</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">Inv</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Olive" title="b1">Inv</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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">  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>)</span><br/>







<span class="kw">thus </span><a NAME="E1:14"/>
 <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>)
  <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#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</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="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) or <font color="Maroon" title="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1"/>
<font color="Maroon" title="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</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="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:14_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
( <font color="Maroon" title="c6">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">xL</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="c5">a</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="c6">xL</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> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">o</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xL</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E4:14_1"/>
<font color="Maroon" title="c6">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <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="#E3:14_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14_1"/>
<font color="Maroon" title="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</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="c5">a</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="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</font>) or <font color="Maroon" title="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:14"/>
<font color="Maroon" title="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><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>)</span>,<font color="Maroon" title="c4">Inv</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="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:14"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
( <font color="Maroon" title="c6">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <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> &amp; <font color="Maroon" title="c6">xL</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="c5">a</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="c6">xL</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> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">o</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xL</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:14"/>
<font color="Maroon" title="c5">a</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="c1">o</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</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/>


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