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<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>  implies  not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:122"/>
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</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">  not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> </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="E3:122"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c4">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="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font>
 <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>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">c</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/>
<a NAME="E5:122"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <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="c6">k</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) as   <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D10" onmouseover="rs('xxreal_0/D10')" onmouseout="rh()">XXREAL_0:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">l</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) as   <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:122"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="xxreal_0.html#D9" onmouseover="rs('xxreal_0/D9')" onmouseout="rh()">XXREAL_0:def 9</a></span>;<br/>
<a NAME="E8:122"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T41" onmouseover="rs('newton03/T41')" onmouseout="rh()">NEWTON03:41</a></span>;<br/>
<a NAME="E9:122"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D1" onmouseover="rs('abian/D1')" onmouseout="rh()">ABIAN:def 1</a></span>;<br/>
<a NAME="E10:122"/><span class="kw">then </span>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton02.html#T2" onmouseover="rs('newton02/T2')" onmouseout="rh()">NEWTON02:2</a></span>;<br/>
<a NAME="E11:122"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_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="E1:122_1"/>
( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> or <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T2" target="_self" onmouseover="rs('newton05/T2')" onmouseout="rh()">ABS</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1"><span class="kw">suppose </span></a><a NAME="E1:122_1_1"/><span class="lab"><font color="Green" title="E49">B1</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</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">  not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> </span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E50">B2</font></span>: <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T5" target="_self" onmouseover="rs('newton05/T5')" onmouseout="rh()">LABS</a></span>
<br/>.= 
<font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T83" target="_self" onmouseover="rs('newton05/T83')" onmouseout="rh()">MMD</a></span>
;<br/><a NAME="E3:122_1_1"/>
 not 4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">l</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E2:122_1_1"><span class="lab"><font color="Green" title="E50">B2</font></span></a>, <a class="ref" href="newton03.html#T20" onmouseover="rs('newton03/T20')" onmouseout="rh()">NEWTON03:20</a></span>;<br/><span class="kw">hence </span><a NAME="E4:122_1_1"/>
 not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122_1_1"><span class="lab"><font color="Green" title="E49">B1</font></span></a>, <a class="txt" href="#E8:122"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="newton05.html#T84" target="_self" onmouseover="rs('newton05/T84')" onmouseout="rh()">A4I</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="122_1_2"><span class="kw">suppose </span></a><a NAME="E1:122_1_2"/><span class="lab"><font color="Green" title="E49">B1</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</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">  not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> </span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E50">B2</font></span>: <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T4" target="_self" onmouseover="rs('newton05/T4')" onmouseout="rh()">LmABS</a></span>
<br/>.= 
<font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T83" target="_self" onmouseover="rs('newton05/T83')" onmouseout="rh()">MMD</a></span>
;<br/><a NAME="E3:122_1_2"/>
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">l</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122_1_2"><span class="lab"><font color="Green" title="E49">B1</font></span></a>, <a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="newton03.html#T41" onmouseover="rs('newton03/T41')" onmouseout="rh()">NEWTON03:41</a>, <a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E48">A5</font></span></a></span>;<br/><a NAME="E4:122_1_2"/><span class="kw">then </span>
 not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T20" onmouseover="rs('newton03/T20')" onmouseout="rh()">NEWTON03:20</a></span>;<br/><span class="kw">hence </span><a NAME="E5:122_1_2"/>
 not <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> is  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122_1_2"><span class="lab"><font color="Green" title="E50">B2</font></span></a>, <a class="ref" href="newton05.html#T84" target="_self" onmouseover="rs('newton05/T84')" onmouseout="rh()">A4I</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>

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