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

<span class="kw">set </span><font color="Maroon" title="c1">G2</font> =  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">A</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</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="c2">D</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">$1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  = 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> $1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</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="c2">D</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">$1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">fx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 st <br/>( <font color="Olive" title="b1">fx</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 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">fx</font>),<font color="Maroon">H<sub>2</sub></font>($1),<font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">fx</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> );<br/>
<a NAME="E1:118"/><span class="lab"><font color="Green" title="E80">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2  ex <font color="Olive" title="b2">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="118_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</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">  ex <font color="Olive" title="b1">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">i2</font> = 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_2.html#D1" onmouseover="rs('ec_pf_2/D1')" onmouseout="rh()">EC_PF_2:def 1</a>, <a class="ref" href="number09.html#T56" onmouseover="rs('number09/T56')" onmouseout="rh()">NUMBER09:56</a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">i2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">i</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c5">i2</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>,<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">i2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">i</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c5">i2</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>]</span><br/>

<span class="kw">thus </span><a NAME="E2:118_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>,<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">i2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">i</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c5">i2</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.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="c4">f</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2<span class="kw"> such that </span><br/><a NAME="E3:118"/><span class="lab"><font color="Green" title="E81">A2</font></span>: 
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S6" onmouseover="rs('pboole/S6')" onmouseout="rh()">PBOOLE:sch 6</a>(<a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E80">A1</font></span></a>);<br/></span>
<a NAME="E4:118"/><span class="lab"><font color="Green" title="E82">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E5:118"/><span class="lab"><font color="Green" title="E83">A4</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="118_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</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="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:118_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</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="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:118_2"/><span class="lab"><font color="Green" title="E84">A5</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E4:118_2"/><span class="lab"><font color="Green" title="E85">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118_2"><span class="lab"><font color="Green" title="E84">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">fx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2<span class="kw"> such that </span><br/><a NAME="E7:118_2"/><span class="lab"><font color="Green" title="E86">A7</font></span>: 
<font color="Maroon" title="c8">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E8:118_2"/><span class="lab"><font color="Green" title="E87">A8</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">fx</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">k</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">fx</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E81">A2</font></span></a></span>;<br/>
<a NAME="E9:118_2"/><span class="lab"><font color="Green" title="E88">A9</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number09.html#T56" onmouseover="rs('number09/T56')" onmouseout="rh()">NUMBER09:56</a></span>;<br/>
<a NAME="E10:118_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E11:118_2"/><span class="kw">then </span>
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118_2"><span class="lab"><font color="Green" title="E86">A7</font></span></a>, <a class="txt" href="#E9:118_2"><span class="lab"><font color="Green" title="E88">A9</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="E12:118_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A10</font></span>: 
<font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/>
<a NAME="E13:118_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A11</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">fx</font>) is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <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">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E14:118_2"/><span class="lab"><font color="Green" title="E91">A12</font></span>: 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">fx</font>) is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <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">by</span> <span class="lab"><a class="txt" href="#E12:118_2"><span class="lab"><font color="Green" title="E89">A10</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E15:118_2"/><span class="lab"><font color="Green" title="E92">A13</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">fx</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></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="complex1.html#T75" onmouseover="rs('complex1/T75')" onmouseout="rh()">COMPLEX1:75</a></span>;<br/>
<a NAME="E16:118_2"/><span class="lab"><font color="Green" title="E93">A14</font></span>: 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">fx</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></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="complex1.html#T75" onmouseover="rs('complex1/T75')" onmouseout="rh()">COMPLEX1:75</a></span>;<br/>
<a NAME="E17:118_2"/><span class="lab"><font color="Green" title="E94">A15</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></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">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></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="pepin.html#T24" onmouseover="rs('pepin/T24')" onmouseout="rh()">PEPIN:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E95">A16</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="c2">D</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span> = 
<font color="Maroon" title="c2">D</font> <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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" onmouseover="rs('newton/T6')" onmouseout="rh()">NEWTON:6</a></span>
;<br/>
<span class="lab"><font color="Green" title="E96">A17</font></span>: <font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">k</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> = 
<font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c7">k</font></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="pepin.html#T24" onmouseover="rs('pepin/T24')" onmouseout="rh()">PEPIN:24</a></span>
<br/>.= 
<font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" onmouseover="rs('newton/T9')" onmouseout="rh()">NEWTON:9</a></span>
;<br/>
<a NAME="E20:118_2"/><span class="lab"><font color="Green" title="E97">A18</font></span>: 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" onmouseover="rs('newton/T6')" onmouseout="rh()">NEWTON:6</a></span>;<br/>
<span class="lab"><font color="Green" title="E98">A19</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span></span>)</span></span>)</span> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118_2"><span class="lab"><font color="Green" title="E86">A7</font></span></a>, <a class="txt" href="#E20:118_2"><span class="lab"><font color="Green" title="E97">A18</font></span></a></span>
<br/>.= 
 <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="c2">D</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:118_2"><span class="lab"><font color="Green" title="E95">A16</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E99">A20</font></span>: <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">fx</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">k</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">D</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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></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="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">k</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:118_2"><span class="lab"><font color="Green" title="E92">A13</font></span></a>, <a class="txt" href="#E17:118_2"><span class="lab"><font color="Green" title="E94">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></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="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">fx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">k</font>) <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:118_2"><span class="lab"><font color="Green" title="E95">A16</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">fx</font>) <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="#E16:118_2"><span class="lab"><font color="Green" title="E93">A14</font></span></a>, <a class="txt" href="#E17:118_2"><span class="lab"><font color="Green" title="E94">A15</font></span></a>, <a class="txt" href="#E19:118_2"><span class="lab"><font color="Green" title="E96">A17</font></span></a>, <a class="txt" href="#E21:118_2"><span class="lab"><font color="Green" title="E98">A19</font></span></a></span>
;<br/>
<a NAME="E23:118_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">fx</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">k</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="#E12:118_2"><span class="lab"><font color="Green" title="E89">A10</font></span></a>, <a class="ref" href="pepin.html#T18" onmouseover="rs('pepin/T18')" onmouseout="rh()">PEPIN:18</a>, <a class="ref" href="nat_5.html#T3" onmouseover="rs('nat_5/T3')" onmouseout="rh()">NAT_5:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:118_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_2"><span class="lab"><font color="Green" title="E85">A6</font></span></a>, <a class="txt" href="#E8:118_2"><span class="lab"><font color="Green" title="E87">A8</font></span></a>, <a class="txt" href="#E13:118_2"><span class="lab"><font color="Green" title="E90">A11</font></span></a>, <a class="txt" href="#E14:118_2"><span class="lab"><font color="Green" title="E91">A12</font></span></a>, <a class="txt" href="#E22:118_2"><span class="lab"><font color="Green" title="E99">A20</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="E6:118"/>
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="118_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</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="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> or <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:118_3"/><span class="lab"><font color="Green" title="E84">A21</font></span>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">and </span><br/><a NAME="E2:118_3"/><span class="lab"><font color="Green" title="E85">A22</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</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="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_3"><span class="lab"><font color="Green" title="E84">A21</font></span></a></span>;<br/>
<a NAME="E4:118_3"/>
(  ex <font color="Olive" title="b1">fx1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 st <br/>( <font color="Olive" title="b1">fx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">fx1</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x1</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">fx1</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> ) &amp;  ex <font color="Olive" title="b1">fx2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="number09.html#K2" title="NUMBER09:func.2">GreaterOrEqualsNumbers</a> 2 st <br/>( <font color="Olive" title="b1">fx2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">x2</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">fx2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">x2</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">fx2</font>)</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:118"><span class="lab"><font color="Green" title="E81">A2</font></span></a></span>;<br/>
<a NAME="E5:118_3"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">x2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_3"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="ref" href="xtuple_0.html#T3" onmouseover="rs('xtuple_0/T3')" onmouseout="rh()">XTUPLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:118_3"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T30" onmouseover="rs('pepin/T30')" onmouseout="rh()">PEPIN:30</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">hence </span><a NAME="E7:118"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118"><span class="lab"><font color="Green" title="E82">A3</font></span></a>, <a class="txt" href="#E5:118"><span class="lab"><font color="Green" title="E83">A4</font></span></a>, <a class="ref" href="card_1.html#T59" onmouseover="rs('card_1/T59')" onmouseout="rh()">CARD_1:59</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>
