<?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>, <font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp;  not <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:24"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" 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>  or  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:24"/>
 not <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a9</font> = <font color="Maroon" title="c1">a</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E5:24"/>
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1"/>
 not <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">b9</font> = <font color="Maroon" title="c2">b</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:24_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">b9</font> = <font color="Maroon" title="c2">b</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E7:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c9</font> = <font color="Maroon" title="c3">c</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E9:24"/>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">c9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D1" title="ABIAN:def.1">ABIAN:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E11:24"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c6">c9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/>
<a NAME="E12:24"/>
<font color="Maroon" title="c3">c</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E13:24"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T16" title="SQUARE_1:th.16">SQUARE_1:16</a></span>;<br/>
<a NAME="E14:24"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E15:24"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">m9</font> = <font color="Maroon" title="c7">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">n9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E18:24"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c6">c9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D2" title="ABIAN:def.2">ABIAN:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">k9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E20:24"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c5">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D2" title="ABIAN:def.2">ABIAN:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A6</font></span>: <font color="Maroon" title="c9">n9</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">m9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">k9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:24"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>
;<br/>
<a NAME="E22:24"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c9">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">m9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/>
<a NAME="E23:24"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c9">n9</font>,<font color="Maroon" title="c8">m9</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font> be   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pythtrip.html#D2" target="_self" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">n9</font> or  not <font color="Maroon" title="c11">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">m9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_3"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">n9</font>
 <span class="kw">and </span><br/><a NAME="E2:24_3"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">m9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">p</font> as   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:24_3"/>
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:24"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E1:24_3"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></span>;<br/>
<a NAME="E5:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E6:24_3"/>
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">m9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="int_2.html#T10" title="INT_2:th.10">INT_2:10</a></span>;<br/>
<a NAME="E7:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">n9</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">m9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_3"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="wsierp_1.html#T4" title="WSIERP_1:th.4">WSIERP_1:4</a></span>;<br/>
<a NAME="E8:24_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E9:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T10" title="INT_2:th.10">INT_2:10</a></span>;<br/>
<a NAME="E10:24_3"/>
<font color="Maroon" title="c2">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/>
<a NAME="E11:24_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E5:24_3"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="wsierp_1.html#T4" title="WSIERP_1:th.4">WSIERP_1:4</a></span>;<br/>
<a NAME="E12:24_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:24_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E7:24_3"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="pythtrip.html#D2" target="_self" title="PYTHTRIP:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:24"/><span class="kw">then </span>
<font color="Maroon" title="c9">n9</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="#E21:24"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="pythtrip.html#T1" target="_self" title="PYTHTRIP:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E26:24"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c9">n9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></span>;<br/>
<a NAME="E27:24"/>
<font color="Maroon" title="c8">m9</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="#E23:24"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E21:24"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="pythtrip.html#T1" target="_self" title="PYTHTRIP:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E29:24"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c8">m9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c11">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">m</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^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="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^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="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> )</span><br/>

<a NAME="E31:24"/>
<font color="Maroon" title="c9">n9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">m9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a></span>;<br/>
<a NAME="E32:24"/><span class="kw">then </span>
<font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="xreal_1.html#T49" title="XREAL_1:th.49">XREAL_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:24"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T16" title="SQUARE_1:th.16">SQUARE_1:16</a></span>; <a class="txt" 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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^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="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E34:24"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^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="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E23">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> )</span><br/>

<font color="Maroon" title="c2">b</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  = 
<span class="p1">(<span class="default">2 <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:24"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E21:24"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="square_1.html#T9" title="SQUARE_1:th.9">SQUARE_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 

;<br/>
<span class="kw">hence </span><a NAME="E36:24"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pythtrip.html#T5" target="_self" title="PYTHTRIP:th.5">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E37:24"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E23">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
