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<span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">a</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</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">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/>
<a NAME="E2:116_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#D1" onmouseover="rs('newton03/D1')" onmouseout="rh()">NEWTON03:def 1</a></span>;<br/>
<a NAME="E3:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#D7" onmouseover="rs('newton03/D7')" onmouseout="rh()">NEWTON03:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:116_1_1"/><span class="lab"><font color="Green" title="E43">A0</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E6:116_1_1"/>
 not <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116_1_1"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <a class="ref" href="newton03.html#T76" onmouseover="rs('newton03/T76')" onmouseout="rh()">NEWTON03:76</a></span>;<br/>
<a NAME="E7:116_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" onmouseover="rs('int_2/D2')" onmouseout="rh()">INT_2:def 2</a></span>;<br/>
<a NAME="E8:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">p</font> <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="pepin.html#T2" onmouseover="rs('pepin/T2')" onmouseout="rh()">PEPIN:2</a></span>;<br/>
<a NAME="E9:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span> <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="wsierp_1.html#T10" onmouseover="rs('wsierp_1/T10')" onmouseout="rh()">WSIERP_1:10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:116_1_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 not <font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E11:116_1_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 not 2 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="116_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_1_1_1"/>
2 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> contradiction</span><br/>

<a NAME="E2:116_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></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>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:116_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:116_1_1"><span class="lab"><font color="Green" title="E45">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>
<a NAME="E12:116_1_1"/>
 for <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds  not <font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <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="116_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <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="c6">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:116_1_1_2"/><span class="lab"><font color="Green" title="E47">B1</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E4:116_1_1_2"/>
<font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="#E3:116_1_1_2"><span class="lab"><font color="Green" title="E47">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="E5:116_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T14" onmouseover="rs('newton03/T14')" onmouseout="rh()">NEWTON03:14</a>, <a class="txt" href="#E9:116_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E11:116_1_1"><span class="lab"><font color="Green" title="E46">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>
<span class="kw">hence </span><a NAME="E13:116_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116_1_1"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <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>
