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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</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">r</font> is  <a href="surrealn.html#V1" title="SURREALN:attr.1">dyadic-like</a>  iff  ex <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b2">n</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:27"/>
( <font color="Maroon" title="c1">r</font> is  <a href="surrealn.html#V1" title="SURREALN:attr.1">dyadic-like</a>  implies  ex <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b2">n</font></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"> (  ex <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b2">n</font></span>)</span> implies <font color="Maroon" title="c1">r</font> is  <a href="surrealn.html#V1" title="SURREALN:attr.1">dyadic-like</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:27_1"/>
<font color="Maroon" title="c1">r</font> is  <a href="surrealn.html#V1" title="SURREALN:attr.1">dyadic-like</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">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b2">n</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font>
 ;<br/>
<span class="kw">take </span>
 <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">r</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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">r</font></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="Olive" title="b1">n</font></span>)</span></span><br/>

<span class="kw">take </span>
<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" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">r</font></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></span><br/>

<span class="kw">thus </span><a NAME="E4:27_1"/>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">r</font></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="rat_1.html#T15" onmouseover="rs('rat_1/T15')" onmouseout="rh()">RAT_1:15</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">given </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><a NAME="E3:27"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c1">r</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>
 ; <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">r</font> is  <a href="surrealn.html#V1" title="SURREALN:attr.1">dyadic-like</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">w</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:27"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">w</font> &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="rat_1.html#T27" onmouseover="rs('rat_1/T27')" onmouseout="rh()">RAT_1:27</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:27"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T28" onmouseover="rs('int_2/T28')" onmouseout="rh()">INT_2:28</a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="int_1.html#D3" onmouseover="rs('int_1/D3')" onmouseout="rh()">INT_1:def 3</a>, <a class="ref" href="pepin.html#T34" onmouseover="rs('pepin/T34')" onmouseout="rh()">PEPIN:34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">nt</font> = <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">t</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<a NAME="E9:27"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">nt</font>
 
;<br/>
<a NAME="E10:27"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c6">nt</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c5">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="newton.html#T8" onmouseover="rs('newton/T8')" onmouseout="rh()">NEWTON:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:27"/>
<font color="Maroon" title="c1">r</font> is  <a href="surrealn.html#V1" title="SURREALN:attr.1">dyadic-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T5" onmouseover="rs('xcmplx_1/T5')" onmouseout="rh()">XCMPLX_1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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