<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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"> ( 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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> implies ( 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> &amp; 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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 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"> ( 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> &amp; 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font> )</span><br/>

<a NAME="E2:56"/><span class="lab"><font color="Green" title="E42">A2</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E3:56"/><span class="lab"><font color="Green" title="E43">A3</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E4:56"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>;<br/>
<a NAME="E5:56"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
2 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>;<br/>
<a NAME="E6:56"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
4 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>;<br/>
<a NAME="E7:56"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T12" onmouseover="rs('int_1/T12')" onmouseout="rh()">INT_1:12</a></span>;<br/>
<a NAME="E8:56"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> , <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <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#R2" title="INT_1:pred.2">are_congruent_mod</a> 7
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>;<br/>
<a NAME="E9:56"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> , <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <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#R2" title="INT_1:pred.2">are_congruent_mod</a> 7
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:56"/>
(  not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> or  not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</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" onclick="hsNdiv(this)" href="javascript:()" title="56_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="E1:56_1"/><span class="kw">then </span>
(  not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> or  not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1"/>
 not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</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 class="add"><a NAME="E2:56_1_1"/><span class="kw">then </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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T48" target="_self" onmouseover="rs('number02/T48')" onmouseout="rh()">Th48</a></span>;<br/><a NAME="E3:56_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> ,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> ,2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> ,4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>;<br/><a NAME="E4:56_1_1"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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#K6" title="XCMPLX_0:func.6">-</a> 4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="int_1.html#T17" onmouseover="rs('int_1/T17')" onmouseout="rh()">INT_1:17</a></span>;<br/><a NAME="E5:56_1_1"/><span class="kw">then </span>
( <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#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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>,<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#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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>,<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#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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>,<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#K3" title="XCMPLX_0:func.3">*</a> 4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 ;<br/><a NAME="E6:56_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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> 7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="int_1.html#T16" onmouseover="rs('int_1/T16')" onmouseout="rh()">INT_1:16</a></span>;<br/><a NAME="E7:56_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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_6.html#T12" onmouseover="rs('nat_6/T12')" onmouseout="rh()">NAT_6:12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:56_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T49" target="_self" onmouseover="rs('number02/T49')" onmouseout="rh()">Th49</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="56_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_2"/>
 not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</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 class="add"><a NAME="E2:56_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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T48" target="_self" onmouseover="rs('number02/T48')" onmouseout="rh()">Th48</a></span>;<br/><a NAME="E3:56_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> ,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> ,2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <font color="Maroon" title="c2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> ,4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>;<br/><a NAME="E4:56_1_2"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E9:56"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="int_1.html#T17" onmouseover="rs('int_1/T17')" onmouseout="rh()">INT_1:17</a></span>;<br/><a NAME="E5:56_1_2"/><span class="kw">then </span>
( <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#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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></span>)</span>,<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#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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></span>)</span>,<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#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<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></span>)</span>,<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#K3" title="XCMPLX_0:func.3">*</a> 4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 ;<br/><a NAME="E6:56_1_2"/><span class="kw">then </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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> 7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 or <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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 7 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 7 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="int_1.html#T16" onmouseover="rs('int_1/T16')" onmouseout="rh()">INT_1:16</a></span>;<br/><a NAME="E7:56_1_2"/><span class="kw">then </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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_6.html#T12" onmouseover="rs('nat_6/T12')" onmouseout="rh()">NAT_6:12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:56_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T49" target="_self" onmouseover="rs('number02/T49')" onmouseout="rh()">Th49</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>
