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<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">Y</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>  st <font color="Olive" title="b2">Y</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Olive" title="b1">X</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Olive" title="b3">Inv</font> is <font color="Olive" title="b1">X</font> <a href="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</a>  holds <br/> <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Olive" title="b2">Y</font>,<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">Inv</font>) 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">X</font>, <font color="Maroon" title="c3">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">  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="Maroon" title="c3">Y</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Maroon" title="c2">X</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Olive" title="b1">Inv</font> is <font color="Maroon" title="c2">X</font> <a href="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</a>  holds <br/> <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">Inv</font>) is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> </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"> ( <font color="Maroon" title="c3">Y</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Maroon" title="c2">X</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Maroon" title="c4">Inv</font> is <font color="Maroon" title="c2">X</font> <a href="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</a>  implies  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">Inv</font>) is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
( <font color="Maroon" title="c3">Y</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Maroon" title="c2">X</font> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp; <font color="Maroon" title="c4">Inv</font> is <font color="Maroon" title="c2">X</font> <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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">Inv</font>) is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">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="c3">Y</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">Inv</font>) or <font color="Maroon" title="c5">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:12"/>
<font color="Maroon" title="c5">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="c3">Y</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">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">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="c6">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
( <font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c5">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="c6">l</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">Inv</font>) )
 <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="c7">l</font> = <font color="Maroon" title="c6">l</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="#E1:12"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>;<br/>
<a NAME="E6:12"/>
 <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c7">l</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">Inv</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#T4" target="_self" onmouseover="rs('surreali/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12"/>
<font color="Maroon" title="c5">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="#E4:12"><span class="lab"><font color="Green" title="E11">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/>


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