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

<span class="kw">set </span><font color="Maroon" title="c2">A</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span> ;<br/>
<span class="kw">thus </span><a NAME="E1:148"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span>  or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:148_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:148_1"/><span class="lab"><font color="Green" title="E95">A1</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:148_1"/><span class="lab"><font color="Green" title="E96">A2</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>
 ;<br/>
<a NAME="E5:148_1"/><span class="lab"><font color="Green" title="E97">A3</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/>
<a NAME="E6:148_1"/><span class="lab"><font color="Green" title="E98">A4</font></span>: 
4 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_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="E1:148_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_1"/><span class="lab"><font color="Green" title="E99">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E6:148_1"><span class="lab"><font color="Green" title="E98">A4</font></span></a>, <a class="ref" href="square_1.html#T40" onmouseover="rs('square_1/T40')" onmouseout="rh()">SQUARE_1:40</a></span>;<br/><span class="kw">hence </span><a NAME="E3:148_1_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E1:148_1_1_1"><span class="lab"><font color="Green" title="E99">A5</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</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="148_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2"/><span class="lab"><font color="Green" title="E99">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">d</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>;<br/><a NAME="E2:148_1_1_2"/><span class="lab"><font color="Green" title="E100">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/><a NAME="E3:148_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</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="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a></span>;<br/><a NAME="E4:148_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A9</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a></span>;<br/><a NAME="E5:148_1_1_2"/><span class="lab"><font color="Green" title="E103">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" onmouseover="rs('int_2/D2')" onmouseout="rh()">INT_2:def 2</a></span>;<br/><a NAME="E6:148_1_1_2"/><span class="lab"><font color="Green" title="E104">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" onmouseover="rs('int_2/D2')" onmouseout="rh()">INT_2:def 2</a></span>;<br/><a NAME="E7:148_1_1_2"/><span class="lab"><font color="Green" title="E105">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</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="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:148_1_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/><a NAME="E8:148_1_1_2"/><span class="lab"><font color="Green" title="E106">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1_1_2"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E6:148_1_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="ref" href="int_5.html#T1" onmouseover="rs('int_5/T1')" onmouseout="rh()">INT_5:1</a></span>;<br/><a NAME="E9:148_1_1_2"/><span class="lab"><font color="Green" title="E107">A14</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><a NAME="E10:148_1_1_2"/><span class="lab"><font color="Green" title="E108">A15</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  holds  not <font color="Olive" title="b1">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c7">p</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><a NAME="E2:148_1_1_2_1"/><span class="lab"><font color="Green" title="E109">A16</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</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"> contradiction</span><br/>

<a NAME="E3:148_1_1_2_1"/><span class="lab"><font color="Green" title="E110">A17</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:148_1_1_2"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="txt" href="#E2:148_1_1_2_1"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E4:148_1_1_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</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="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:148_1_1_2"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E2:148_1_1_2_1"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E5:148_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A18</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E2:148_1_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="ref" href="number14.html#T6" target="_self" onmouseover="rs('number14/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E6:148_1_1_2_1"/><span class="lab"><font color="Green" title="E112">A19</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:148_1_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E2:148_1_1_2_1"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E7:148_1_1_2_1"/><span class="lab"><font color="Green" title="E113">A20</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 or  ex <font color="Olive" title="b1">z</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; <font color="Olive" title="b1">z</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_1"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="ref" href="number14.html#T8" target="_self" onmouseover="rs('number14/T8')" onmouseout="rh()">Th8</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/>
<a NAME="E8:148_1_1_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1_1_2"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E2:148_1_1_2_1"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E9:148_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:148_1_1_2_1"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="ref" href="wsierp_1.html#T4" onmouseover="rs('wsierp_1/T4')" onmouseout="rh()">WSIERP_1:4</a></span>;<br/>
<a NAME="E10:148_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1_1_2_1"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="ref" href="wsierp_1.html#T4" onmouseover="rs('wsierp_1/T4')" onmouseout="rh()">WSIERP_1:4</a></span>;<br/>
<a NAME="E11:148_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:148_1_1_2_1"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="ref" href="int_5.html#T1" onmouseover="rs('int_5/T1')" onmouseout="rh()">INT_5:1</a></span>;<br/>
<a NAME="E12:148_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 or <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number05.html#T21" onmouseover="rs('number05/T21')" onmouseout="rh()">NUMBER05:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:148_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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><span class="kw">consider </span><font color="Maroon" title="c7">E</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c8">b</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:148_1_1_2"/><span class="lab"><font color="Green" title="E109">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="ref" href="nat_6.html#T3" onmouseover="rs('nat_6/T3')" onmouseout="rh()">NAT_6:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E110">A22</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:148_1_1_2_2"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:148_1_1_2_2"/><span class="lab"><font color="Green" title="E111">A23</font></span>: 
<font color="Maroon" title="c9">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E4:148_1_1_2_2"/><span class="lab"><font color="Green" title="E112">A24</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T31" onmouseover="rs('int_2/T31')" onmouseout="rh()">INT_2:31</a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/><a NAME="E5:148_1_1_2_2"/>
<font color="Maroon" title="c9">p</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_2"><span class="lab"><font color="Green" title="E112">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:148_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:148_1_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E3:148_1_1_2_2"><span class="lab"><font color="Green" title="E111">A23</font></span></a>, <a class="txt" href="#E4:148_1_1_2_2"><span class="lab"><font color="Green" title="E112">A24</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</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><span class="kw">consider </span><font color="Maroon" title="c9">m</font>, <font color="Maroon" title="c10">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E15:148_1_1_2"/><span class="lab"><font color="Green" title="E111">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E16:148_1_1_2"/><span class="lab"><font color="Green" title="E112">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">and </span><br/><a NAME="E17:148_1_1_2"/><span class="lab"><font color="Green" title="E113">A27</font></span>: 
<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="ref" href="int_2.html#T23" onmouseover="rs('int_2/T23')" onmouseout="rh()">INT_2:23</a></span>;<br/><a NAME="E18:148_1_1_2"/><span class="lab"><font color="Green" title="E114">A28</font></span>: 
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3"><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="E1:148_1_1_2_3"/>
( 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_1"/><span class="lab"><font color="Green" title="E115">A29</font></span>: 
16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A30</font></span>: 
16 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T1" onmouseover="rs('nat_3/T1')" onmouseout="rh()">NAT_3:1</a></span>;<br/><a NAME="E3:148_1_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</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="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1_1_2"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E6:148_1_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="ref" href="nat_3.html#T1" onmouseover="rs('nat_3/T1')" onmouseout="rh()">NAT_3:1</a></span>;<br/><a NAME="E4:148_1_1_2_3_1"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E2:148_1_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_1"><span class="lab"><font color="Green" title="E116">A30</font></span></a>, <a class="txt" href="number14.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="int_2.html#T9" onmouseover="rs('int_2/T9')" onmouseout="rh()">INT_2:9</a></span>;<br/><a NAME="E5:148_1_1_2_3_1"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="number14.html#T101" target="_self" onmouseover="rs('number14/T101')" onmouseout="rh()">Th101</a></span>;<br/><a NAME="E6:148_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A31</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="nat_3.html#T1" onmouseover="rs('nat_3/T1')" onmouseout="rh()">NAT_3:1</a></span>;<br/><a NAME="E7:148_1_1_2_3_1"/>
16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:148_1_1_2"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_1"><span class="lab"><font color="Green" title="E115">A29</font></span></a>, <a class="ref" href="int_2.html#T9" onmouseover="rs('int_2/T9')" onmouseout="rh()">INT_2:9</a></span>;<br/><a NAME="E8:148_1_1_2_3_1"/><span class="kw">then </span>
16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:148_1_1_2_3_1"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="ref" href="int_5.html#T1" onmouseover="rs('int_5/T1')" onmouseout="rh()">INT_5:1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:148_1_1_2_3_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</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="148_1_1_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2"/>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2"/><span class="kw">then </span>
 not 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a>, <a class="ref" href="polyeq_5.html#T3" onmouseover="rs('polyeq_5/T3')" onmouseout="rh()">POLYEQ_5:3</a></span>;<br/><a NAME="E3:148_1_1_2_3_2"/><span class="kw">then </span>
3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</a></span>;<br/><a NAME="E4:148_1_1_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">E</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:148_1_1_2_3_2"/><span class="kw">then </span>
 not  not <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_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="E1:148_1_1_2_3_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1"/>
<font color="Maroon" title="c7">E</font> <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:148_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E116">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="number14.html#T103" target="_self" onmouseover="rs('number14/T103')" onmouseout="rh()">Th103</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E6:148_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E117">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E115">A32</font></span></a>, <a class="ref" href="number14.html#T103" target="_self" onmouseover="rs('number14/T103')" onmouseout="rh()">Th103</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E116">A33</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E117">A34</font></span></a>, <a class="ref" href="newton.html#T7" onmouseover="rs('newton/T7')" onmouseout="rh()">NEWTON:7</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A35</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="newton03.html#T13" onmouseover="rs('newton03/T13')" onmouseout="rh()">NEWTON03:13</a></span>;<br/><a NAME="E9:148_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E119">A36</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E116">A33</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E117">A34</font></span></a></span>;<br/><a NAME="E10:148_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E120">A37</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><a NAME="E11:148_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E121">A38</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_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="E1:148_1_1_2_3_2_1_1_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8 <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="txt" href="#E9:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E119">A36</font></span></a></span>;<br/><a NAME="E3:148_1_1_2_3_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:148_1_1_2_3_2_1_1_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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="148_1_1_2_3_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_2"/><span class="lab"><font color="Green" title="E122">A39</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_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="E1:148_1_1_2_3_2_1_1_1_2_1"/>
( <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E118">A35</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E123">A40</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="lab"><font color="Green" title="E124">A41</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E116">A33</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E117">A34</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E118">A35</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E120">A37</font></span></a>, <a class="txt" href="#E11:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E121">A38</font></span></a></span>
;<br/><a NAME="E3:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E125">A42</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_2_2"/>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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/>

<a NAME="E2:148_1_1_2_3_2_1_1_1_2_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E124">A41</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c5">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  = 
<span class="p1">(<span class="default">7 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E118">A35</font></span></a>, <a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 7</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 7</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 121
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>
<br/>.= 
464

;<br/>
<span class="kw">hence </span><a NAME="E4:148_1_1_2_3_2_1_1_1_2_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E137"><span class="lab"><font color="Green" title="E94">Lm13</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><a NAME="E4:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E126">A43</font></span>: 
3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E123">A40</font></span></a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A44</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E6:148_1_1_2_3_2_1_1_1_2_1_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E123">A40</font></span></a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E124">A41</font></span></a>, <a class="txt" href="#E5:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E127">A44</font></span></a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E9:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E125">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E10:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A45</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A46</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E126">A43</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E12:148_1_1_2_3_2_1_1_1_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E128">A45</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E13:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E129">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E14:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E124">A41</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E15:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 ;<br/><a NAME="E16:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E123">A40</font></span></a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E17:148_1_1_2_3_2_1_1_1_2_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T9" onmouseover="rs('int_1/T9')" onmouseout="rh()">INT_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E18:148_1_1_2_3_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2"><span class="lab"><font color="Green" title="E122">A39</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_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="E1:148_1_1_2_3_2_1_1_1_2_1_3_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or ( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T133" onmouseover="rs('xreal_1/T133')" onmouseout="rh()">XREAL_1:133</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E123">A47</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E124">A48</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E3:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E125">A49</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="power.html#T2" onmouseover="rs('power/T2')" onmouseout="rh()">POWER:2</a></span>;<br/><a NAME="E4:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E126">A50</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span></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> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E123">A47</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E6:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A51</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E124">A48</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></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> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E9:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E120">A37</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E126">A50</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E127">A51</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E10:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 ;<br/><a NAME="E11:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">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"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">b</font></span>)</span> <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"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E12:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A53</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E125">A49</font></span></a>, <a class="txt" href="#E9:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E128">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E13:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E130">A54</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E116">A33</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E117">A34</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E118">A35</font></span></a></span>;<br/><a NAME="E14:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 8 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E15:148_1_1_2_3_2_1_1_1_2_1_3_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E129">A53</font></span></a>, <a class="txt" href="#E13:148_1_1_2_3_2_1_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E130">A54</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E123">A55</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E124">A56</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E3:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E116">A33</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E117">A34</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E118">A35</font></span></a></span>;<br/><a NAME="E4:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 8 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E120">A37</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E123">A55</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E124">A56</font></span></a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E6:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E124">A56</font></span></a>, <a class="txt" href="number14.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a>, <a class="ref" href="prepower.html#T9" onmouseover="rs('prepower/T9')" onmouseout="rh()">PREPOWER:9</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A57</font></span>: 
1 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E116">A33</font></span></a>, <a class="txt" href="#E6:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E117">A34</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E118">A35</font></span></a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E123">A55</font></span></a>, <a class="ref" href="int_1.html#T52" onmouseover="rs('int_1/T52')" onmouseout="rh()">INT_1:52</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_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="E1:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 or <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:148_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E120">A37</font></span></a>, <a class="txt" href="#E7:148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E125">A57</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A58</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T52" onmouseover="rs('int_1/T52')" onmouseout="rh()">INT_1:52</a></span>;<br/><a NAME="E3:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="number14.html#T104" target="_self" onmouseover="rs('number14/T104')" onmouseout="rh()">Th104</a></span>;<br/><a NAME="E4:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A59</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span></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> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <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> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"><span class="lab"><font color="Green" title="E126">A58</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 8</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> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"><span class="lab"><font color="Green" title="E127">A59</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 12</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 7
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E8:148_1_1_2_3_2_1_1_1_2_1_3_1_2_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:148_1_1_2_3_2_1_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E125">A57</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><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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E115">A60</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A61</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a></span>;<br/><a NAME="E3:148_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E117">A62</font></span>: 
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E5:148_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E2:148_1_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E7:148_1_1_2"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E115">A60</font></span></a>, <a class="ref" href="number14.html#T6" target="_self" onmouseover="rs('number14/T6')" onmouseout="rh()">Th6</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/><a NAME="E4:148_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E118">A63</font></span>: 
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a>, <a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E115">A60</font></span></a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_2"/>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 ;<br/><a NAME="E6:148_1_1_2_3_2_1_2"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E118">A63</font></span></a>, <a class="ref" href="wsierp_1.html#T4" onmouseover="rs('wsierp_1/T4')" onmouseout="rh()">WSIERP_1:4</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_2"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E117">A62</font></span></a>, <a class="ref" href="wsierp_1.html#T4" onmouseover="rs('wsierp_1/T4')" onmouseout="rh()">WSIERP_1:4</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A64</font></span>: 
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E116">A61</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E120">A65</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_3_2_1_2_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) or ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; contradiction ) or ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; contradiction ) or ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_2_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="E1:148_1_1_2_3_2_1_2_1_1"/>
( ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) or ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) or ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) or ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c10">n</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="148_1_1_2_3_2_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:148_1_1_2_3_2_1_2_1_1_1"/>
( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</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"> ( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_2_1_1_1"/>
( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:148_1_1_2_3_2_1_2_1_1_2"/>
( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c10">n</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"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E119">A64</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:148_1_1_2_3_2_1_2_1_1_3"/>
( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c10">n</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"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_2_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E119">A64</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_2_1_1_4"><span class="kw">case </span></a><a NAME="E1:148_1_1_2_3_2_1_2_1_1_4"/>
( <font color="Maroon" title="c9">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c10">n</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"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_2_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="ref" href="newton05.html#T35" onmouseover="rs('newton05/T35')" onmouseout="rh()">NEWTON05:35</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><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E11:148_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E121">A66</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">n1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E13:148_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E122">A67</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E120">A65</font></span></a>, <a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">u</font> = 8 <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="c11">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">n1</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E123">A68</font></span>: <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 = 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a>, <a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E115">A60</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default">8 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E121">A66</font></span></a>, <a class="txt" href="#E13:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E122">A67</font></span></a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">8 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">8 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">n1</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">4 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">4 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T71" onmouseover="rs('nat_d/T71')" onmouseout="rh()">NAT_D:71</a></span>
<br/>.= 
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="E16:148_1_1_2_3_2_1_2"/><span class="kw">then </span>
 not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E123">A68</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a></span>;<br/><span class="kw">hence </span><a NAME="E17:148_1_1_2_3_2_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E117">A62</font></span></a>, <a class="txt" href="number14.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="newton02.html#T15" onmouseover="rs('newton02/T15')" onmouseout="rh()">NEWTON02:15</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="148_1_1_2_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_3"/>
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A69</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a>, <a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A70</font></span>: <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 = 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a></span>
<br/>.= 
16 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font></span>)</span>

;<br/><a NAME="E4:148_1_1_2_3_2_1_3"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A71</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="number14.html#T100" target="_self" onmouseover="rs('number14/T100')" onmouseout="rh()">Th100</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">w</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E7:148_1_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E118">A72</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font>
 ;<br/><a NAME="E8:148_1_1_2_3_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">x</font> <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">16 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 4</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E118">A72</font></span></a></span>;<br/><a NAME="E9:148_1_1_2_3_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <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="c11">w</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E116">A70</font></span></a>, <a class="txt" href="#E9:148_1_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a></span>;<br/><a NAME="E10:148_1_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A73</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font>
 ;<br/><a NAME="E11:148_1_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E120">A74</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:148_1_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E115">A69</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a></span>;<br/><a NAME="E12:148_1_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E121">A75</font></span>: 
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 ;<br/><a NAME="E13:148_1_1_2_3_2_1_3"/>
( ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) or ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_3_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:148_1_1_2_3_2_1_3_2"/>
(  not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> or <font color="Maroon" title="c10">n</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"> ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_3_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="E1:148_1_1_2_3_2_1_3_2_1"/><span class="kw">then </span>
( (  not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) or (  not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) or ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c10">n</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="148_1_1_2_3_2_1_3_2_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_3_2_1_1"/><span class="lab"><font color="Green" title="E122">A76</font></span>: 
 not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_3_2_1_1"/><span class="lab"><font color="Green" title="E123">A77</font></span>: 
<font color="Maroon" title="c10">n</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"> ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E3:148_1_1_2_3_2_1_3_2_1_1"/>
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="txt" href="#E10:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E119">A73</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E122">A76</font></span></a>, <a class="txt" href="number14.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="number14.html#T102" target="_self" onmouseover="rs('number14/T102')" onmouseout="rh()">Th102</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</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"> <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><span class="kw">hence </span><a NAME="E4:148_1_1_2_3_2_1_3_2_1_1"/>
<font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E121">A75</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E123">A77</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_3_2_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_3_2_1_2"/><span class="lab"><font color="Green" title="E122">A78</font></span>: 
 not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_3_2_1_2"/><span class="lab"><font color="Green" title="E123">A79</font></span>: 
<font color="Maroon" title="c10">n</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"> ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E3:148_1_1_2_3_2_1_3_2_1_2"/>
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="txt" href="#E10:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E119">A73</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_3_2_1_2"><span class="lab"><font color="Green" title="E122">A78</font></span></a>, <a class="txt" href="number14.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="number14.html#T102" target="_self" onmouseover="rs('number14/T102')" onmouseout="rh()">Th102</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</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"> <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><span class="kw">thus </span><a NAME="E4:148_1_1_2_3_2_1_3_2_1_2"/>
<font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_3_2_1_2"><span class="lab"><font color="Green" title="E123">A79</font></span></a>, <a class="ref" href="newton05.html#T35" onmouseover="rs('newton05/T35')" onmouseout="rh()">NEWTON05:35</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="148_1_1_2_3_2_1_3_2_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_3_2_1_3"/><span class="lab"><font color="Green" title="E122">A80</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_3_2_1_3"/><span class="lab"><font color="Green" title="E123">A81</font></span>: 
<font color="Maroon" title="c10">n</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"> ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><a NAME="E3:148_1_1_2_3_2_1_3_2_1_3"/>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E121">A75</font></span></a>, <a class="txt" href="#E1:148_1_1_2_3_2_1_3_2_1_3"><span class="lab"><font color="Green" title="E122">A80</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/><a NAME="E4:148_1_1_2_3_2_1_3_2_1_3"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_3_2_1_3"><span class="lab"><font color="Green" title="E123">A81</font></span></a>, <a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="ref" href="int_2.html#D2" onmouseover="rs('int_2/D2')" onmouseout="rh()">INT_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:148_1_1_2_3_2_1_3_2_1_3"/>
( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</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="148_1_1_2_3_2_1_3_3"><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="E1:148_1_1_2_3_2_1_3_3"/><span class="kw">then </span>
( ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c10">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) or ( 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">m</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="148_1_1_2_3_2_1_3_3_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_3_3_1"/><span class="lab"><font color="Green" title="E122">A82</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_3_3_1"/><span class="lab"><font color="Green" title="E123">A83</font></span>: 
<font color="Maroon" title="c10">n</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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c12">n1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:148_1_1_2_3_2_1_3_3_1"/><span class="lab"><font color="Green" title="E124">A84</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_3_3_1"><span class="lab"><font color="Green" title="E123">A83</font></span></a>, <a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_3_3_1"/><span class="lab"><font color="Green" title="E125">A85</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font>
 ;<br/><a NAME="E6:148_1_1_2_3_2_1_3_3_1"/>
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E120">A74</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_3_3_1"><span class="lab"><font color="Green" title="E124">A84</font></span></a>, <a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E115">A69</font></span></a>, <a class="txt" href="#E5:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E117">A71</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_3_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A86</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1_1_2_3_2_1_3_3_1"><span class="lab"><font color="Green" title="E125">A85</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_3_3_1"/>
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_3_3_1"><span class="lab"><font color="Green" title="E122">A82</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:148_1_1_2_3_2_1_3_3_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:148_1_1_2_3_2_1_3_3_1"><span class="lab"><font color="Green" title="E126">A86</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</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="148_1_1_2_3_2_1_3_3_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:148_1_1_2_3_2_1_3_3_2"/><span class="lab"><font color="Green" title="E122">A87</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">and </span><br/><a NAME="E2:148_1_1_2_3_2_1_3_3_2"/><span class="lab"><font color="Green" title="E123">A88</font></span>: 
<font color="Maroon" title="c9">m</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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c12">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:148_1_1_2_3_2_1_3_3_2"/><span class="lab"><font color="Green" title="E124">A89</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_3_3_2"><span class="lab"><font color="Green" title="E123">A88</font></span></a>, <a class="ref" href="abian.html#T1" onmouseover="rs('abian/T1')" onmouseout="rh()">ABIAN:1</a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_3_3_2"/><span class="lab"><font color="Green" title="E125">A90</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">m1</font>
 ;<br/><a NAME="E6:148_1_1_2_3_2_1_3_3_2"/>
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E120">A74</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_3_3_2"><span class="lab"><font color="Green" title="E124">A89</font></span></a>, <a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E115">A69</font></span></a>, <a class="txt" href="#E5:148_1_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E117">A71</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_3_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A91</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:148_1_1_2_3_2_1_3_3_2"><span class="lab"><font color="Green" title="E125">A90</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_3_3_2"/>
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_3_3_2"><span class="lab"><font color="Green" title="E122">A87</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:148_1_1_2_3_2_1_3_3_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:148_1_1_2_3_2_1_3_3_2"><span class="lab"><font color="Green" title="E126">A91</font></span></a>, <a class="ref" href="int_2.html#T1" onmouseover="rs('int_2/T1')" onmouseout="rh()">INT_2:1</a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</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="148_1_1_2_3_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_4"/>
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A92</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4</span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2"><span class="lab"><font color="Green" title="E109">A21</font></span></a>, <a class="txt" href="#E13:148_1_1_2"><span class="lab"><font color="Green" title="E110">A22</font></span></a>, <a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A93</font></span>: <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 = 
<span class="p1">(<span class="default">8 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">8 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a></span>
<br/>.= 
64 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font></span>)</span>

;<br/><span class="lab"><font color="Green" title="E117">A94</font></span>: <font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:148_1_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E3:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E116">A93</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>
;<br/><a NAME="E5:148_1_1_2_3_2_1_4"/>
16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 16 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 4
 ;<br/><a NAME="E6:148_1_1_2_3_2_1_4"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E116">A93</font></span></a>, <a class="txt" href="number14.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A95</font></span>: 
4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="number14.html#T100" target="_self" onmouseover="rs('number14/T100')" onmouseout="rh()">Th100</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E9:148_1_1_2_3_2_1_4"/><span class="lab"><font color="Green" title="E119">A96</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E117">A94</font></span></a>, <a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="number14.html#T103" target="_self" onmouseover="rs('number14/T103')" onmouseout="rh()">Th103</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E11:148_1_1_2_3_2_1_4"/><span class="lab"><font color="Green" title="E120">A97</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E117">A94</font></span></a>, <a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="txt" href="#E7:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E118">A95</font></span></a>, <a class="ref" href="number14.html#T103" target="_self" onmouseover="rs('number14/T103')" onmouseout="rh()">Th103</a></span>;<br/><a NAME="E12:148_1_1_2_3_2_1_4"/><span class="lab"><font color="Green" title="E121">A98</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E3:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E116">A93</font></span></a>, <a class="txt" href="#E9:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E119">A96</font></span></a></span>;<br/><a NAME="E13:148_1_1_2_3_2_1_4"/><span class="lab"><font color="Green" title="E122">A99</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E3:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E116">A93</font></span></a>, <a class="txt" href="#E11:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E120">A97</font></span></a></span>;<br/><a NAME="E14:148_1_1_2_3_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E117">A94</font></span></a>, <a class="txt" href="#E9:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E119">A96</font></span></a>, <a class="txt" href="#E11:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E120">A97</font></span></a>, <a class="ref" href="newton.html#T7" onmouseover="rs('newton/T7')" onmouseout="rh()">NEWTON:7</a></span>;<br/><a NAME="E15:148_1_1_2_3_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:148_1_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="newton03.html#T13" onmouseover="rs('newton03/T13')" onmouseout="rh()">NEWTON03:13</a></span>;<br/><a NAME="E16:148_1_1_2_3_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A100</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:148_1_1_2"><span class="lab"><font color="Green" title="E111">A25</font></span></a>, <a class="txt" href="#E16:148_1_1_2"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E115">A92</font></span></a>, <a class="txt" href="#E9:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E119">A96</font></span></a>, <a class="txt" href="#E11:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E120">A97</font></span></a></span>;<br/><a NAME="E17:148_1_1_2_3_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A101</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" onmouseover="rs('newton/T11')" onmouseout="rh()">NEWTON:11</a></span>;<br/><a NAME="E18:148_1_1_2_3_2_1_4"/><span class="lab"><font color="Green" title="E125">A102</font></span>: 
( <font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> &amp; <font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_5.html#T2" onmouseover="rs('polyeq_5/T2')" onmouseout="rh()">POLYEQ_5:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E126">A103</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_3_2_1_4_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:148_1_1_2_3_2_1_4_3"/>
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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/><a NAME="E2:148_1_1_2_3_2_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <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="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:148_1_1_2_3_2_1_4_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E124">A101</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_4_4"><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="E1:148_1_1_2_3_2_1_4_4"/>
( <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_4_4_1"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_4_4_1"/>
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:148_1_1_2_3_2_1_4_4_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E121">A98</font></span></a>, <a class="txt" href="#E13:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E122">A99</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_2_3_2_1_4_4_2"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_4_4_2"/><span class="lab"><font color="Green" title="E127">A104</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E3:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A105</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E123">A100</font></span></a>, <a class="txt" href="#E18:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E125">A102</font></span></a></span>;<br/><a NAME="E4:148_1_1_2_3_2_1_4_4_2"/><span class="lab"><font color="Green" title="E129">A106</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E127">A104</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="148_1_1_2_3_2_1_4_4_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">a</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:148_1_1_2_3_2_1_4_4_2_1"/>
<font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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/><a NAME="E2:148_1_1_2_3_2_1_4_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E124">A101</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:148_1_1_2_3_2_1_4_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:148_1_1_2_3_2_1_4_4_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E128">A105</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E129">A106</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><a NAME="E6:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A107</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E9:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E129">A106</font></span></a>, <a class="txt" href="#E7:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E130">A107</font></span></a>, <a class="ref" href="xreal_1.html#T97" onmouseover="rs('xreal_1/T97')" onmouseout="rh()">XREAL_1:97</a></span>;<br/><a NAME="E10:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E123">A100</font></span></a>, <a class="txt" href="#E18:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E125">A102</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E11:148_1_1_2_3_2_1_4_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A108</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E12:148_1_1_2_3_2_1_4_4_2"/>
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E127">A104</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E13:148_1_1_2_3_2_1_4_4_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:148_1_1_2_3_2_1_4_4_2"><span class="lab"><font color="Green" title="E131">A108</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="148_1_1_2_3_2_1_4_4_3"><span class="kw">suppose </span></a><a NAME="E1:148_1_1_2_3_2_1_4_4_3"/><span class="lab"><font color="Green" title="E127">A109</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:148_1_1_2_3_2_1_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A110</font></span>: 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E129">A111</font></span>: <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> = 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p4">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E125">A102</font></span></a></span>
<br/>.= 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><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="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
;<br/><a NAME="E4:148_1_1_2_3_2_1_4_4_3"/><span class="lab"><font color="Green" title="E130">A112</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><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="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><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="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E126">A103</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E5:148_1_1_2_3_2_1_4_4_3"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E6:148_1_1_2_3_2_1_4_4_3"/><span class="kw">then </span>
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><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="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E127">A109</font></span></a>, <a class="ref" href="topreal6.html#T2" onmouseover="rs('topreal6/T2')" onmouseout="rh()">TOPREAL6:2</a></span>;<br/><a NAME="E7:148_1_1_2_3_2_1_4_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A113</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><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="c12">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">a</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E128">A110</font></span></a>, <a class="txt" href="#E4:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E130">A112</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:148_1_1_2_3_2_1_4_4_3"/><span class="lab"><font color="Green" title="E132">A114</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E127">A109</font></span></a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><a NAME="E9:148_1_1_2_3_2_1_4_4_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E10:148_1_1_2_3_2_1_4_4_3"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:148_1_1_2_3_2_1_4"><span class="lab"><font color="Green" title="E123">A100</font></span></a>, <a class="txt" href="#E3:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E129">A111</font></span></a>, <a class="txt" href="#E2:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E128">A110</font></span></a>, <a class="txt" href="#E7:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E131">A113</font></span></a>, <a class="txt" href="#E8:148_1_1_2_3_2_1_4_4_3"><span class="lab"><font color="Green" title="E132">A114</font></span></a>, <a class="ref" href="number14.html#T2" target="_self" onmouseover="rs('number14/T2')" onmouseout="rh()">Th2</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:148"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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"> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span> </span><br/>

<a NAME="E3:148"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A115</font></span>: 
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,4</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E4:148"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 3</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:148"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> : <font color="Olive" title="b2">y</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148"><span class="lab"><font color="Green" title="E95">A115</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>
