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<span class="kw">reconsider </span><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">x</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="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">s</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> holds <br/> ( <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c3">s</font> iff ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or ( <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp;  <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="Olive" title="b1">z</font> ) ) )
 <span class="kw">and </span><br/><a NAME="E5:30_1_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> holds <br/> ( <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c3">s</font> iff ( <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp;  <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="Olive" title="b1">z</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="surreali.html#E26"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E6:30_1_1"/>
 <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="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c3">s</font> as   <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">s</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">  for <font color="Olive" title="b1">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> holds <br/> ( (  not <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">s</font> or <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  ) ) &amp; ( ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  ) ) implies <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">s</font> ) &amp; ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">s</font> implies ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  ) ) &amp; ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  implies <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">s</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:30_1_1"/>
 for <font color="Olive" title="b1">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> holds <br/> ( (  not <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">s</font> or <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  ) ) &amp; ( ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  ) ) implies <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">s</font> ) &amp; ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">s</font> implies ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  ) ) &amp; ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  implies <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">s</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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