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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be   <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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:123"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )</span><br/>

<a NAME="E2:123"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number05.html#T8" onmouseover="rs('number05/T8')" onmouseout="rh()">NUMBER05:8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:123"/>
(  not <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  not <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  not <font color="Maroon" title="c3">c</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_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="E1:123_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_1"/>
<font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_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="E1:123_1_1_1"/>
( <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_1_1_1"/>
<font color="Maroon" title="c2">b</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/><div class="add"><a NAME="E2:123_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_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="E1:123_1_1_1_1_1"/>
( <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E2:123_1_1_1_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="123_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">c</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/><div class="add"><a NAME="E2:123_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E2:123_1_1_1_1"><span class="lab"><font color="Green" title="E85">A4</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_1_1_2"/>
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_2_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="E1:123_1_1_1_2_1"/>
( <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E2:123_1_1_1_2"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="123_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">c</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/><div class="add"><a NAME="E2:123_1_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E2:123_1_1_1_2"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_2"/>
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_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="E1:123_1_2_1"/>
( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_2_1_1"/>
<font color="Maroon" title="c1">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/><div class="add"><a NAME="E2:123_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_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="E1:123_1_2_1_1_1"/>
( <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_2"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E2:123_1_2_1_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="123_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">c</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/><div class="add"><a NAME="E2:123_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_2"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E2:123_1_2_1_1"><span class="lab"><font color="Green" title="E85">A7</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_2_1_2"/>
<font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_1_2_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="E1:123_1_2_1_2_1"/>
( <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_2_1_2_1_1"/>
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_2_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_2"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E2:123_1_2_1_2"><span class="lab"><font color="Green" title="E85">A8</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="123_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_2_1_2_1_2"/>
<font color="Maroon" title="c3">c</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/><div class="add"><a NAME="E2:123_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_2"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E2:123_1_2_1_2"><span class="lab"><font color="Green" title="E85">A8</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3"><span class="kw">suppose </span></a><a NAME="E1:123_1_3"/>
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_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="E1:123_1_3_1"/>
( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_3_1_1"/>
<font color="Maroon" title="c1">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/><div class="add"><a NAME="E2:123_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_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="E1:123_1_3_1_1_1"/>
( <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_3_1_1_1_1"/>
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_3_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_3_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_3"><span class="lab"><font color="Green" title="E84">A9</font></span></a>, <a class="txt" href="#E2:123_1_3_1_1"><span class="lab"><font color="Green" title="E85">A10</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="123_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_3_1_1_1_2"/>
<font color="Maroon" title="c2">b</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/><div class="add"><a NAME="E2:123_1_3_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_3_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_3"><span class="lab"><font color="Green" title="E84">A9</font></span></a>, <a class="txt" href="#E2:123_1_3_1_1"><span class="lab"><font color="Green" title="E85">A10</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_3_1_2"/>
<font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_1_2_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="E1:123_1_3_1_2_1"/>
( <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_3_1_2_1_1"/>
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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/><div class="add"><a NAME="E2:123_1_3_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T4" onmouseover="rs('pythtrip/T4')" onmouseout="rh()">PYTHTRIP:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_3_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_3"><span class="lab"><font color="Green" title="E84">A9</font></span></a>, <a class="txt" href="#E2:123_1_3_1_2"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="123_1_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_3_1_2_1_2"/>
<font color="Maroon" title="c2">b</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/><div class="add"><a NAME="E2:123_1_3_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T3" onmouseover="rs('pythtrip/T3')" onmouseout="rh()">PYTHTRIP:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_3_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:123"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:123_1_3"><span class="lab"><font color="Green" title="E84">A9</font></span></a>, <a class="txt" href="#E2:123_1_3_1_2"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
