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

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="pepin.html#K2">^2</a> 
 ;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<b>assume </b><a NAME="E3:24"/>
not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E5:24"/>
<font color="Maroon">b</font> is <a href="abian.html#V1">even</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
not <font color="Maroon">b</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:24_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E7:24"/>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="pepin.html#K2">^2</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:24"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n'</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E12:24"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="abian.html#D1">ABIAN:def 1</a></i>;<br/>
<a NAME="E13:24"/>
<font color="Maroon">c</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:24"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>
<a NAME="E15:24"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E16:24"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E18:24"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:24"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k'</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/>
<a NAME="E21:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:24"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:24"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n'</font>,<font color="Maroon">m'</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="int_2.html#V1">prime</a> <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="pythtrip.html#D2" target="_self">PYTHTRIP:def 2</a><br/>

<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="int_2.html#V1">prime</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:24_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n'</font>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/>
<a NAME="E4:24_2"/>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">m'</font>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/>
<a NAME="E5:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m'</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T14">INT_2:14</a></i>;<br/>
<a NAME="E6:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n'</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wsierp_1.html#T9">WSIERP_1:9</a></i>;<br/>
<a NAME="E7:24_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/>
<a NAME="E8:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E9:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/>
<a NAME="E10:24_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T14">INT_2:14</a></i>;<br/>
<a NAME="E11:24_2"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T55">NAT_1:55</a></i>;<br/>
<a NAME="E12:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E13:24_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/>
<a NAME="E14:24_2"/>
<font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="wsierp_1.html#T9">WSIERP_1:9</a></i>;<br/>
<a NAME="E16:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/>
<a NAME="E17:24_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/>
<b>hence </b><a NAME="E18:24_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E51</font></i>: <font color="Maroon">n'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="square_1.html#K3">^2</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">k'</font> <a href="pepin.html#K2">^2</a> 
<b>by </b><i/>
;<br/>
<a NAME="E25:24"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">n'</font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">m'</font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E27:24"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E29:24"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>take </b>
<font color="Maroon">n</font>
;<br/>

<a NAME="E31:24"/>
<font color="Maroon">n'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E32:24"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a>  )
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E33:24"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>

<b>thus </b><a NAME="E34:24"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>

<font color="Maroon">b</font> <a href="pepin.html#K2">^2</a>  = 
<span class="p1">(<span class="default">2 <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="pepin.html#K2">^2</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="square_1.html#T68">SQUARE_1:68</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="pepin.html#K2">^2</a> 

;<br/>
<b>hence </b><a NAME="E36:24"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E37:24"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
