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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> </span><br/><div class="add"><a NAME="E2:117_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN: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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">a</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E4:117_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c3">a</font>,2 <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#D5" onmouseover="rs('newton03/D5')" onmouseout="rh()">NEWTON03:def 5</a></span>;<br/><a NAME="E5:117_1_1_1_1"/>
 not <span class="p1">(<span class="default">2 <a href="power.html#K1" title="POWER:func.1">-root</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T28" onmouseover="rs('int_2/T28')" onmouseout="rh()">INT_2:28</a></span>;<br/><a NAME="E6:117_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 not 2 <a href="power.html#K1" title="POWER:func.1">-root</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ;<br/><a NAME="E7:117_1_1_1_1"/>
 for <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:117_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</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="c4">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:117_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">B1</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 ;<br/>
<a NAME="E4:117_1_1_1_1_1"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">B1</font></span></a>, <a class="ref" href="newton.html#T81" onmouseover="rs('newton/T81')" onmouseout="rh()">NEWTON:81</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:117_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_1_1_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E6:117_1_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="newton03.html#T14" onmouseover="rs('newton03/T14')" onmouseout="rh()">NEWTON03:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:117_1_1_1_1"/>
 not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#D3" onmouseover="rs('pythtrip/D3')" onmouseout="rh()">PYTHTRIP:def 3</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="117_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_1_1_1_2"/>
<font color="Maroon" title="c2">a</font> <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 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 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:117_1_1_1_2"/>
 not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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