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<span class="kw">let </span><font color="Maroon" title="c1">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="c1">d</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> or  ex <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">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="Olive" title="b2">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="Olive" title="b1">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c1">d</font> is  not   <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">  ex <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">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="Olive" title="b2">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="Olive" title="b1">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E2:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 not <font color="Maroon" title="c1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:40"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c3">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/>
<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><font color="Maroon" title="c1">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">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E5:40"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="surrealn.html#T21" target="_self" onmouseover="rs('surrealn/T21')" onmouseout="rh()">Th21</a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E31">A3</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<a NAME="E7:40"/>
<font color="Maroon" title="c1">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="c4">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="surrealn.html#D4" target="_self" onmouseover="rs('surrealn/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E8:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 ex <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>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:40"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">m</font>] &amp; (  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" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" onmouseover="rs('nat_1/S5')" onmouseout="rh()">NAT_1:sch 5</a>(<a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E32">A4</font></span></a>);<br/></span>
<a NAME="E11:40"/>
 not <font color="Maroon" title="c1">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="c5">m</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c1">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="c5">m</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="E2:40_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="surrealn.html#T21" target="_self" onmouseover="rs('surrealn/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">m1</font> = <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<a NAME="E4:40_1"/>
<font color="Maroon" title="c6">m1</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="c6">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:40_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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="E12:40"/><span class="kw">then </span>
<font color="Maroon" title="c1">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="c5">m</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="c5">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E33">A5</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/>
<a NAME="E13:40"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">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="Olive" title="b1">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="c5">m</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/>
<span class="kw">hence </span><a NAME="E14:40"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">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="Olive" title="b2">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="Olive" title="b1">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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"> verum</span><br/>


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