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<span class="kw">let </span><font color="Maroon" title="c1">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" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> )</span><br/>

<a NAME="E2:72"/><span class="lab"><font color="Green" title="E65">AA</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_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="E1:72_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4000 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4000 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><span class="kw">suppose </span></a><a NAME="E1:72_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4000
 ; <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="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> )</span><br/><div class="add"><a NAME="E2:72_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4001
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:72_1_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E65">AA</font></span></a>, <a class="txt" href="nat_4.html#E47"><span class="lab"><font color="Green" title="E44">Lm7</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="72_1_2"><span class="kw">suppose </span></a><a NAME="E1:72_1_2"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4000
 ; <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="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c2">m</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c3">X1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c4">X2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c5">X3</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> ;<br/><span class="kw">assume </span><a NAME="E2:72_1_2"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  holds <br/>(  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> or  not <font color="Olive" title="b1">p</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="c1">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"> contradiction</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1_2_1"><span class="kw">now </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">  not <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:72_1_2_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:72_1_2_1"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:72_1_2_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  st <br/>( <font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</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="c1">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E3:72_1_2_1"><span class="lab"><font color="Green" title="E68">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:72_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1_2"><span class="lab"><font color="Green" title="E67">A3</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><a NAME="E4:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E65">AA</font></span></a>, <a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="nat_4.html#E66"><span class="lab"><font color="Green" title="E62">Lm10</font></span></a>, <a class="ref" href="finseq_3.html#T43" onmouseover="rs('finseq_3/T43')" onmouseout="rh()">FINSEQ_3:43</a>, <a class="ref" href="rvsum_1.html#T94" onmouseover="rs('rvsum_1/T94')" onmouseout="rh()">RVSUM_1:94</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:72_1_2"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<span class="p1">(<span class="default">4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T6" target="_self" onmouseover="rs('nat_4/T6')" onmouseout="rh()">Th6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> ;<br/><a NAME="E6:72_1_2"/><span class="lab"><font color="Green" title="E70">A7</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:72_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E8:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E9:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A8</font></span>: 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T46" target="_self" onmouseover="rs('nat_4/T46')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">mm</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">mm</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> as    <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">by</span> <span class="lab"><a class="ref" href="int_1.html#T53" onmouseover="rs('int_1/T53')" onmouseout="rh()">INT_1:53</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">XX</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">mm</font>;<br/><div><span class="lab"><font color="Green" title="E72">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1_2_2"><span class="kw">now </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">  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:72_1_2_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </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"> contradiction</span><br/><a NAME="E2:72_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 )
 ;<br/><span class="kw">hence </span><a NAME="E3:72_1_2_2"/>
contradiction
 ; <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="E12:72_1_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span>
 <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="E13:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A10</font></span>: 
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="power.html#K2" title="POWER:func.2">to_power</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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" onmouseover="rs('power/T39')" onmouseout="rh()">POWER:39</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1_2_3"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b2">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span>  implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:72_1_2_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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="E3:72_1_2_3"/><span class="lab"><font color="Green" title="E74">A11</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E4:72_1_2_3"/><span class="lab"><font color="Green" title="E75">A12</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E5:72_1_2_3"/><span class="lab"><font color="Green" title="E76">A13</font></span>: 
<font color="Maroon" title="c11">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3
 <span class="kw">and </span><br/><a NAME="E6:72_1_2_3"/><span class="lab"><font color="Green" title="E77">A14</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E7:72_1_2_3"/>
<font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E65">AA</font></span></a>, <a class="txt" href="#E6:72_1_2"><span class="lab"><font color="Green" title="E70">A7</font></span></a>, <a class="txt" href="#E4:72_1_2_3"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="txt" href="#E5:72_1_2_3"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="txt" href="nat_4.html#E58"><span class="lab"><font color="Green" title="E54">Lm9</font></span></a></span>;<br/><a NAME="E8:72_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <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>;<br/><a NAME="E9:72_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_1_2_3"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/><a NAME="E10:72_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_1_2_3"><span class="lab"><font color="Green" title="E74">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:72_1_2_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72_1_2_3"><span class="lab"><font color="Green" title="E76">A13</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><a NAME="E15:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A15</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> 
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1_2_4"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b2">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">mm</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span>  implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">mm</font> )</span><br/><span class="kw">assume </span><a NAME="E1:72_1_2_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">mm</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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="E3:72_1_2_4"/><span class="lab"><font color="Green" title="E75">A16</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E4:72_1_2_4"/><span class="lab"><font color="Green" title="E76">A17</font></span>: 
<font color="Maroon" title="c11">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3
 ;<br/><a NAME="E5:72_1_2_4"/><span class="lab"><font color="Green" title="E77">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><a NAME="E6:72_1_2_4"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" onmouseover="rs('int_1/D7')" onmouseout="rh()">INT_1:def 7</a></span>;<br/><a NAME="E7:72_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1_2_4"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:72_1_2_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_1_2_4"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="txt" href="#E5:72_1_2_4"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</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="E17:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A19</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">mm</font>
 ;<br/><a NAME="E18:72_1_2"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E19:72_1_2"/><span class="kw">then </span>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 </span> } </span> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:72_1_2"><span class="lab"><font color="Green" title="E75">A19</font></span></a>, <a class="txt" href="#E11:72_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E15:72_1_2"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="nat_4.html#T43" target="_self" onmouseover="rs('nat_4/T43')" onmouseout="rh()">Th42</a></span>;<br/><a NAME="E20:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A20</font></span>: 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : (  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:72_1_2"><span class="lab"><font color="Green" title="E71">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E21:72_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E22:72_1_2"/><span class="kw">then </span>
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> </span>)</span> <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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E65">AA</font></span></a>, <a class="txt" href="nat_4.html#E67"><span class="lab"><font color="Green" title="E63">Lm11</font></span></a></span>;<br/><a NAME="E23:72_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="c1">n</font></span>)</span> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:72_1_2"><span class="lab"><font color="Green" title="E76">A20</font></span></a>, <a class="txt" href="#E4:72_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="xreal_1.html#T66" onmouseover="rs('xreal_1/T66')" onmouseout="rh()">XREAL_1:66</a></span>;<br/><a NAME="E24:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A21</font></span>: 
<span class="p1">(<span class="default">4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72_1_2"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E25:72_1_2"/><span class="lab"><font color="Green" title="E78">A22</font></span>: 
4 <a href="power.html#K2" title="POWER:func.2">to_power</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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><a NAME="E26:72_1_2"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><a NAME="E27:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72_1_2"><span class="lab"><font color="Green" title="E73">A10</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E28:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:72_1_2"><span class="lab"><font color="Green" title="E77">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E29:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E30:72_1_2"/><span class="kw">then </span>
( 4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> &amp; 4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T41" onmouseover="rs('power/T41')" onmouseout="rh()">POWER:41</a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E31:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p4">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:72_1_2"><span class="lab"><font color="Green" title="E78">A22</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E32:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:72_1_2"><span class="lab"><font color="Green" title="E78">A22</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E33:72_1_2"/><span class="kw">then </span>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T29" onmouseover="rs('power/T29')" onmouseout="rh()">POWER:29</a></span>;<br/><a NAME="E34:72_1_2"/><span class="kw">then </span>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" onmouseover="rs('power/T25')" onmouseout="rh()">POWER:25</a></span>;<br/><a NAME="E35:72_1_2"/><span class="kw">then </span>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <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="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T27" onmouseover="rs('power/T27')" onmouseout="rh()">POWER:27</a></span>;<br/><a NAME="E36:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A23</font></span>: 
( 4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> or 4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <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="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E37:72_1_2"/>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><a NAME="E38:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:72_1_2"><span class="lab"><font color="Green" title="E79">A23</font></span></a>, <a class="ref" href="power.html#T37" onmouseover="rs('power/T37')" onmouseout="rh()">POWER:37</a></span>;<br/><a NAME="E39:72_1_2"/><span class="kw">then </span>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a></span>;<br/><a NAME="E40:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A24</font></span>: 
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c1">n</font> <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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">2n</font> = 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="lab"><font color="Green" title="E81">A25</font></span>: <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 6 = 
<span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c10">2n</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 6
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" onmouseover="rs('power/T41')" onmouseout="rh()">POWER:41</a></span>
<br/>.= 
<font color="Maroon" title="c10">2n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T4" onmouseover="rs('comptrig/T4')" onmouseout="rh()">COMPTRIG:4</a></span>
;<br/>2 <a href="power.html#K2" title="POWER:func.2">to_power</a> 2 = 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" onmouseover="rs('power/T41')" onmouseout="rh()">POWER:41</a></span>
<br/>.= 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,2</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T61" onmouseover="rs('finseq_2/T61')" onmouseout="rh()">FINSEQ_2:61</a></span>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T99" onmouseover="rs('rvsum_1/T99')" onmouseout="rh()">RVSUM_1:99</a></span>
;<br/><a NAME="E44:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A26</font></span>: 
( 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; 4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a>, <a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">n1</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><a NAME="E45:72_1_2"/>
<span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D6" onmouseover="rs('int_1/D6')" onmouseout="rh()">INT_1:def 6</a></span>;<br/><a NAME="E46:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A27</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E47:72_1_2"/>
6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6 <a href="power.html#K1" title="POWER:func.1">-root</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T17" onmouseover="rs('power/T17')" onmouseout="rh()">POWER:17</a></span>;<br/><a NAME="E48:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A28</font></span>: 
6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T5" onmouseover="rs('power/T5')" onmouseout="rh()">POWER:5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">n1</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> as    <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">by</span> <span class="lab"><a class="ref" href="int_1.html#T53" onmouseover="rs('int_1/T53')" onmouseout="rh()">INT_1:53</a></span>;<br/><a NAME="E50:72_1_2"/>
<font color="Maroon" title="c12">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D6" onmouseover="rs('int_1/D6')" onmouseout="rh()">INT_1:def 6</a></span>;<br/><a NAME="E51:72_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> or <font color="Maroon" title="c12">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <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="E52:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A29</font></span>: 
2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c12">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" onmouseover="rs('power/T39')" onmouseout="rh()">POWER:39</a></span>;<br/><a NAME="E53:72_1_2"/>
<font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c12">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T85" onmouseover="rs('newton/T85')" onmouseout="rh()">NEWTON:85</a></span>;<br/><a NAME="E54:72_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c12">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" onmouseover="rs('power/T41')" onmouseout="rh()">POWER:41</a></span>;<br/><a NAME="E55:72_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:72_1_2"><span class="lab"><font color="Green" title="E85">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E56:72_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> )
 <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="E57:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 6 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 6
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T37" onmouseover="rs('power/T37')" onmouseout="rh()">POWER:37</a></span>;<br/><a NAME="E58:72_1_2"/>
<span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 6 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:72_1_2"><span class="lab"><font color="Green" title="E83">A27</font></span></a>, <a class="txt" href="#E48:72_1_2"><span class="lab"><font color="Green" title="E84">A28</font></span></a>, <a class="ref" href="power.html#T37" onmouseover="rs('power/T37')" onmouseout="rh()">POWER:37</a></span>;<br/><a NAME="E59:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:72_1_2"><span class="lab"><font color="Green" title="E86">A30</font></span></a>, <a class="txt" href="#E42:72_1_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E60:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A31</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a></span>;<br/><a NAME="E61:72_1_2"/>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="square_1.html#T25" onmouseover="rs('square_1/T25')" onmouseout="rh()">SQUARE_1:25</a></span>;<br/><a NAME="E62:72_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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E60:72_1_2"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="ref" href="power.html#T37" onmouseover="rs('power/T37')" onmouseout="rh()">POWER:37</a></span>;<br/><a NAME="E63:72_1_2"/><span class="kw">then </span>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:72_1_2"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E64:72_1_2"/><span class="kw">then </span>
4 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a></span>;<br/><a NAME="E65:72_1_2"/><span class="kw">then </span>
 <a href="power.html#K3" title="POWER:func.3">log</a> (2,<span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K3" title="POWER:func.3">log</a> (2,<span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:72_1_2"><span class="lab"><font color="Green" title="E82">A26</font></span></a>, <a class="ref" href="power.html#T57" onmouseover="rs('power/T57')" onmouseout="rh()">POWER:57</a></span>;<br/><a NAME="E66:72_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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,2)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K3" title="POWER:func.3">log</a> (2,<span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T55" onmouseover="rs('power/T55')" onmouseout="rh()">POWER:55</a></span>;<br/><a NAME="E67:72_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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,2)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,2)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T55" onmouseover="rs('power/T55')" onmouseout="rh()">POWER:55</a></span>;<br/><a NAME="E68:72_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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">log</a> (2,2)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T52" onmouseover="rs('power/T52')" onmouseout="rh()">POWER:52</a></span>;<br/><a NAME="E69:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A32</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 6</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T52" onmouseover="rs('power/T52')" onmouseout="rh()">POWER:52</a></span>;<br/><a NAME="E70:72_1_2"/>
42 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E71:72_1_2"/><span class="kw">then </span>
42 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
 <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="E72:72_1_2"/><span class="kw">then </span>
81 <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="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E73:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A33</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> 81 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T27" onmouseover="rs('square_1/T27')" onmouseout="rh()">SQUARE_1:27</a></span>;<br/><a NAME="E74:72_1_2"/>
81 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 <a href="pepin.html#K1" title="PEPIN:func.1">^2</a> 
 ;<br/><a NAME="E75:72_1_2"/><span class="kw">then </span>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> 81 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T22" onmouseover="rs('square_1/T22')" onmouseout="rh()">SQUARE_1:22</a></span>;<br/><a NAME="E76:72_1_2"/><span class="kw">then </span>
9 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:72_1_2"><span class="lab"><font color="Green" title="E89">A33</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E77:72_1_2"/><span class="kw">then </span>
18 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">18 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">18 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 <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="E78:72_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">18 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">18 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:72_1_2"><span class="lab"><font color="Green" title="E84">A28</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E79:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:72_1_2"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E80:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T32" onmouseover="rs('prepower/T32')" onmouseout="rh()">PREPOWER:32</a></span>;<br/><a NAME="E81:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#D1" onmouseover="rs('power/D1')" onmouseout="rh()">POWER:def 1</a></span>;<br/><a NAME="E82:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T45" onmouseover="rs('power/T45')" onmouseout="rh()">POWER:45</a></span>;<br/><a NAME="E83:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 6</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T45" onmouseover="rs('power/T45')" onmouseout="rh()">POWER:45</a></span>;<br/><a NAME="E84:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A34</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 6</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T27" onmouseover="rs('power/T27')" onmouseout="rh()">POWER:27</a></span>;<br/><a NAME="E85:72_1_2"/><span class="lab"><font color="Green" title="E91">A35</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <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="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><a NAME="E86:72_1_2"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><a NAME="E87:72_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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">20 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:72_1_2"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E88:72_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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:72_1_2"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E89:72_1_2"/><span class="kw">then </span>
<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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" onmouseover="rs('power/T25')" onmouseout="rh()">POWER:25</a></span>;<br/><a NAME="E90:72_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A36</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T29" onmouseover="rs('power/T29')" onmouseout="rh()">POWER:29</a></span>;<br/><a NAME="E91:72_1_2"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/><a NAME="E92:72_1_2"/><span class="kw">then </span>
<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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E90:72_1_2"><span class="lab"><font color="Green" title="E92">A36</font></span></a>, <a class="ref" href="power.html#T37" onmouseover="rs('power/T37')" onmouseout="rh()">POWER:37</a></span>;<br/><a NAME="E93:72_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="c1">n</font></span>)</span> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="power.html#T33" onmouseover="rs('power/T33')" onmouseout="rh()">POWER:33</a></span>;<br/><a NAME="E94:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 20 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" onmouseover="rs('power/T25')" onmouseout="rh()">POWER:25</a></span>;<br/><a NAME="E95:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" onmouseover="rs('power/T27')" onmouseout="rh()">POWER:27</a></span>;<br/><a NAME="E96:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">20 <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 20
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" onmouseover="rs('power/T25')" onmouseout="rh()">POWER:25</a></span>;<br/><a NAME="E97:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 20
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" onmouseover="rs('power/T27')" onmouseout="rh()">POWER:27</a></span>;<br/><a NAME="E98:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">20 <a href="power.html#K2" title="POWER:func.2">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 20</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 20
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" onmouseover="rs('power/T25')" onmouseout="rh()">POWER:25</a></span>;<br/><a NAME="E99:72_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">20 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 20</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 20
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" onmouseover="rs('power/T25')" onmouseout="rh()">POWER:25</a></span>;<br/><a NAME="E100:72_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="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 8000 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><span class="kw">hence </span><a NAME="E101:72_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1_2"><span class="lab"><font color="Green" title="E66">A2</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>
