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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span><br/>

<a NAME="E1:91"/>
( <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> &amp; <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="surrealn.html#E74"><span class="lab"><font color="Green" title="E70">Lm13</font></span></a>, <a class="ref" href="surrealn.html#T46" target="_self" onmouseover="rs('surrealn/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E2:91"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )
 
<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="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E3:91"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D1" target="_self" onmouseover="rs('surrealn/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E4:91"/><span class="kw">then </span>
(  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/>
<a NAME="E5:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A1</font></span>: 
 <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T37" onmouseover="rs('surrealr/T37')" onmouseout="rh()">SURREALR:37</a></span>;<br/>
<a NAME="E6:91"/>
<span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/>
<a NAME="E7:91"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:91"/>
 <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</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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