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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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"> (  not <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  not <font color="Maroon" title="c2">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:65"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
(  not <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  not <font color="Maroon" title="c2">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span><br/>

<a NAME="E2:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  &amp;  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</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#T33" target="_self" onmouseover="rs('surreali/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<a NAME="E3:65"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T54" onmouseover="rs('surrealr/T54')" onmouseout="rh()">SURREALR:54</a>, <a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/>
<a NAME="E4:65"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/>
<a NAME="E5:65"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealr.html#T54" onmouseover="rs('surrealr/T54')" onmouseout="rh()">SURREALR:54</a></span>;<br/>
<a NAME="E6:65"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E7:65"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</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="#E1:65"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="surreali.html#T41" target="_self" onmouseover="rs('surreali/T41')" onmouseout="rh()">Lm3</a>, <a class="ref" href="surreali.html#T42" target="_self" onmouseover="rs('surreali/T42')" onmouseout="rh()">Th41</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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