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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> holds  <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> ) &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">si</font>, <font color="Olive" title="b4">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b3">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Olive" title="b3">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b4">sj</font> ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( (  for <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> $1 holds <br/><a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> ) &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">si</font>, <font color="Olive" title="b4">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b3">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1</span>)</span></span>)</span> &amp; <font color="Olive" title="b4">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1</span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Olive" title="b3">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b4">sj</font> ) ) );<br/>
<a NAME="E1:42"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" onmouseover="rs('newton/T4')" onmouseout="rh()">NEWTON:4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:42_1"/>
 for <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> holds <br/><a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> is   <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">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">si</font>, <font color="Olive" title="b4">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b3">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Olive" title="b3">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b4">sj</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">d</font> be   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</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="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> implies <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c2">d</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T21" target="_self" onmouseover="rs('surrealn/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E3:42_1_1"/>
<a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> <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> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:42_1_1"/>
<a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> is   <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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">si</font>, <font color="Olive" title="b2">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> holds <br/>( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> iff <font color="Olive" title="b1">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b2">sj</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">si</font>, <font color="Maroon" title="c5">sj</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"> ( <font color="Maroon" title="c4">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> implies ( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> iff <font color="Maroon" title="c4">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c5">sj</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
( <font color="Maroon" title="c4">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</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"> ( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> iff <font color="Maroon" title="c4">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c5">sj</font> )</span><br/>

<a NAME="E4:42_1"/>
( <font color="Maroon" title="c4">si</font> <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> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c5">sj</font> <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> <font color="Maroon" title="c3">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42_1"/>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> iff <font color="Maroon" title="c4">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c5">sj</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T9" target="_self" onmouseover="rs('surrealn/T9')" onmouseout="rh()">Th9</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>
<a NAME="E2:42"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">n1</font> = <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<span class="kw">thus </span><a NAME="E2:42_2"/>
 for <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/><a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> is   <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">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">si</font>, <font color="Olive" title="b4">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b3">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Olive" title="b3">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b4">sj</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">d</font> be   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</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="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> implies <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_2_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</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"> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_2_1_1"/>
( <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> or  not <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</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"> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_2_1_1_1"/>
<a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_1_2"/>
 not <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</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"> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a></span><br/><div class="add"><a NAME="E2:42_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:42_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T20" target="_self" onmouseover="rs('surrealn/T20')" onmouseout="rh()">Th20</a></span>;<br/><a NAME="E5:42_2_1_1_2"/>
( <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D4" target="_self" onmouseover="rs('surrealn/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">si</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>, <font color="Maroon" title="c7">si1</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><a NAME="E7:42_2_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:42_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">si1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><a NAME="E9:42_2_1_1_2"/>
<a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">si</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">si1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a>, <a class="txt" href="#E4:42_2_1_1_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_2_1_1_2"/>
<a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_2_1_1_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="surrealo.html#T22" onmouseover="rs('surrealo/T22')" onmouseout="rh()">SURREALO:22</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">si</font>, <font color="Olive" title="b2">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> holds <br/>( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Olive" title="b1">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b2">sj</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">si</font>, <font color="Maroon" title="c7">sj</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"> ( <font color="Maroon" title="c6">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> implies ( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E3:42_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
( <font color="Maroon" title="c6">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</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"> ( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/>

<a NAME="E4:42_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" onmouseover="rs('newton/T6')" onmouseout="rh()">NEWTON:6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_2_2"/>
( <font color="Maroon" title="c4">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c4">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:42_2_2_1"/>
<font color="Maroon" title="c4">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:42_2_2_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T11" onmouseover="rs('abian/T11')" onmouseout="rh()">ABIAN:11</a></span>;<br/><a NAME="E4:42_2_2_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T91" onmouseover="rs('xcmplx_1/T91')" onmouseout="rh()">XCMPLX_1:91</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_2_2_1_1"/>
( <font color="Maroon" title="c5">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c5">j</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_2_2_1_1_1"/>
<font color="Maroon" title="c5">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">j1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:42_2_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T11" onmouseover="rs('abian/T11')" onmouseout="rh()">ABIAN:11</a></span>;<br/><a NAME="E4:42_2_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_1_1_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T91" onmouseover="rs('xcmplx_1/T91')" onmouseout="rh()">XCMPLX_1:91</a></span>;<br/><a NAME="E5:42_2_2_1_1_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2_2_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E4:42_2_2_1_1_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_2_2_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E3:42_2_2_1_1_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_2_2_1_1_2"/>
<font color="Maroon" title="c5">j</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">j1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:42_2_2_1_1_2"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><a NAME="E4:42_2_2_1_1_2"/>
( <font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D4" target="_self" onmouseover="rs('surrealn/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Dn</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>, <font color="Maroon" title="c11">D1n</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><a NAME="E6:42_2_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c7">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">Dn</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">D1n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E7:42_2_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c7">sj</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">sj</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">sj</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c7">sj</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/><span class="kw">thus </span><a NAME="E8:42_2_2_1_1_2"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</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="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> implies <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_2_2_1_1_2_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</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="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font></span><br/>

<a NAME="E2:42_2_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a>, <a class="txt" href="#E3:42_2_2_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E3:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/>
<a NAME="E3:42_2_2_1_1_2_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">Dn</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c6">si</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E4:42_2_2_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:42_2_2_1_1_2_1"/>
<font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a>, <a class="txt" href="#E7:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E40">A17</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/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E9:42_2_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
( <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</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"> contradiction</span><br/><a NAME="E10:42_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E3:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E11:42_2_2_1_1_2"/><span class="kw">then </span>
( 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><a NAME="E12:42_2_2_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E13:42_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">D1n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c6">si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E4:42_2_2_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a></span>;<br/><a NAME="E14:42_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">D1n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c7">sj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E41">A18</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="E15:42_2_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a>, <a class="txt" href="#E7:42_2_2_1_1_2"><span class="lab"><font color="Green" title="E40">A17</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:42_2_2_2"/>
<font color="Maroon" title="c4">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:42_2_2_2"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><a NAME="E4:42_2_2_2"/>
( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D4" target="_self" onmouseover="rs('surrealn/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Sn</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>, <font color="Maroon" title="c10">S1n</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><a NAME="E6:42_2_2_2"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
<font color="Maroon" title="c6">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">Sn</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">S1n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E7:42_2_2_2"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c6">si</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">si</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">si</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c6">si</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_2_2_2_1"/>
( <font color="Maroon" title="c5">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c5">j</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_2_2_2_1_1"/>
<font color="Maroon" title="c5">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:42_2_2_2_1_1"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T11" onmouseover="rs('abian/T11')" onmouseout="rh()">ABIAN:11</a></span>;<br/><a NAME="E4:42_2_2_2_1_1"/><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2_1_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T91" onmouseover="rs('xcmplx_1/T91')" onmouseout="rh()">XCMPLX_1:91</a></span>;<br/><span class="kw">thus </span><a NAME="E5:42_2_2_2_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</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="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> implies <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_2_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_2_2_2_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</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="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font></span><br/>

<a NAME="E2:42_2_2_2_1_1_1"/><span class="kw">then </span>
( 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:42_2_2_2_1_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/>
<a NAME="E3:42_2_2_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E4:42_2_2_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">sj</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c10">S1n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E4:42_2_2_2_1_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42_2_2_2_1_1_1"/>
<font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_2_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E6:42_2_2_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</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">assume </span><a NAME="E6:42_2_2_2_1_1"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
( <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</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"> contradiction</span><br/><a NAME="E7:42_2_2_2_1_1"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:42_2_2_2_1_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E8:42_2_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><a NAME="E9:42_2_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">sj</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c9">Sn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E4:42_2_2_2_1_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a></span>;<br/><a NAME="E10:42_2_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">sj</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c6">si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_2_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E6:42_2_2_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</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="E11:42_2_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_2_2_2_1_1"><span class="lab"><font color="Green" title="E41">A24</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_2_2_2_1_2"/>
<font color="Maroon" title="c5">j</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> iff <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:42_2_2_2_1_2"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><a NAME="E4:42_2_2_2_1_2"/>
( <font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D4" target="_self" onmouseover="rs('surrealn/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Dn</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>, <font color="Maroon" title="c13">D1n</font> = <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><a NAME="E6:42_2_2_2_1_2"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
<font color="Maroon" title="c7">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">Dn</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">D1n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="surrealn.html#D5" target="_self" onmouseover="rs('surrealn/D5')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E7:42_2_2_2_1_2"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c7">sj</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">sj</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">sj</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c7">sj</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/><span class="kw">thus </span><a NAME="E8:42_2_2_2_1_2"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</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="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> implies <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_2_2_2_1_2_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</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="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font></span><br/>

<a NAME="E2:42_2_2_2_1_2_1"/><span class="kw">then </span>
( 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/>
<a NAME="E3:42_2_2_2_1_2_1"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:42_2_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/>
<a NAME="E5:42_2_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">S1n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c12">Dn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/>
<a NAME="E6:42_2_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">S1n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c7">sj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a>, <a class="txt" href="#E7:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E41">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:42_2_2_2_1_2_1"/>
<font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_2_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E6:42_2_2_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</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">assume </span><a NAME="E9:42_2_2_2_1_2"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
( <font color="Maroon" title="c6">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c7">sj</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</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"> contradiction</span><br/><a NAME="E10:42_2_2_2_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E11:42_2_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E12:42_2_2_2_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E9:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="surrealo.html#T3" onmouseover="rs('surrealo/T3')" onmouseout="rh()">SURREALO:3</a></span>;<br/><a NAME="E13:42_2_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E3:42_2_2_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E11:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E14:42_2_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><a NAME="E15:42_2_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">D1n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c9">Sn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/><a NAME="E16:42_2_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">D1n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c6">si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_2_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E6:42_2_2_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a></span>;<br/><a NAME="E17:42_2_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">sj</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c6">si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E6:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</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="E18:42_2_2_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_2_2_2_1_2"><span class="lab"><font color="Green" title="E42">A28</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:42"/><span class="lab"><font color="Green" title="E33">A30</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E32">A4</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E4:42"/>
 for <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> holds  <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> is   <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">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">si</font>, <font color="Olive" title="b4">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b3">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Olive" title="b3">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b4">sj</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">d</font> be   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</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="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:42_3"/><span class="lab"><font color="Green" title="E34">A31</font></span>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T18" target="_self" onmouseover="rs('surrealn/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E3:42_3"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surrealn.html#K3" title="SURREALN:func.3">DYADIC</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_3"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="ref" href="surrealn.html#D4" target="_self" onmouseover="rs('surrealn/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:42_3"/>
<a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">d</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E33">A30</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:42"/>
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">si</font>, <font color="Olive" title="b4">sj</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b3">si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">sj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> iff <font color="Olive" title="b3">si</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Olive" title="b4">sj</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E33">A30</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/>


</div>
