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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/>

<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b4">p</font> where <font color="Olive" title="b4">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is  <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</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"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is  <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ] <span class="kw">means </span>1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ] <span class="kw">means </span>$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/>
<a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 for <font color="Olive" title="b1">q</font> being  <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">q</font>] )
 
;<br/>
<a NAME="E2:2_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is  <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is  <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] </span> } </span> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="hilb10_3.html#S2" onmouseover="rs('hilb10_3/S2')" onmouseout="rh()">HILB10_3:sch 2</a>(<a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:2_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is  <font color="Maroon" title="c2">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_3.html#T20" onmouseover="rs('hilb10_3/T20')" onmouseout="rh()">HILB10_3:20</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="E2:2"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b4">p</font> where <font color="Olive" title="b4">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>$4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 <a href="newton.html#K2" title="NEWTON:func.2">!</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/>
<a NAME="E1:2_2"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">i3</font>, <font color="Olive" title="b5">i4</font>, <font color="Olive" title="b6">i5</font>, <font color="Olive" title="b7">i6</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b8">p</font> where <font color="Olive" title="b8">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i5</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i6</font>] </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_4.html#T32" onmouseover="rs('hilb10_4/T32')" onmouseout="rh()">HILB10_4:32</a></span>;<br/>
<a NAME="E2:2_2"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">i3</font>, <font color="Olive" title="b5">i4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b6">p</font> where <font color="Olive" title="b6">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font> </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<a NAME="E3:2_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">i3</font>, <font color="Olive" title="b5">i4</font>, <font color="Olive" title="b6">i5</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b7">p</font> where <font color="Olive" title="b7">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i5</font>),<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i5</font>] </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="hilb10_3.html#S4" onmouseover="rs('hilb10_3/S4')" onmouseout="rh()">HILB10_3:sch 4</a>(<a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:2_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b4">p</font> where <font color="Olive" title="b4">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>$4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <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">(<span class="default">$2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> ;<br/>
<a NAME="E3:2"/><span class="lab"><font color="Green" title="E5">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">i3</font>, <font color="Olive" title="b5">i4</font>, <font color="Olive" title="b6">i5</font>, <font color="Olive" title="b7">i6</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b8">p</font> where <font color="Olive" title="b8">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i5</font>,<font color="Olive" title="b8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i6</font>] </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_3.html#T15" onmouseover="rs('hilb10_3/T15')" onmouseout="rh()">HILB10_3:15</a></span>;<br/>
<a NAME="E4:2"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">i3</font>, <font color="Olive" title="b5">i4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b6">p</font> where <font color="Olive" title="b6">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font> </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A3</font></span></a></span>;<br/>
<a NAME="E5:2"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">i3</font>, <font color="Olive" title="b5">i4</font>, <font color="Olive" title="b6">i5</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">n</font> holds  <span class="p1"> { <span class="default"> <font color="Olive" title="b7">p</font> where <font color="Olive" title="b7">p</font> is  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i2</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i5</font>),<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i3</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i4</font>,<font color="Olive" title="b7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i5</font>] </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="hilb10_3.html#S4" onmouseover="rs('hilb10_3/S4')" onmouseout="rh()">HILB10_3:sch 4</a>(<a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E6">A7</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</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"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</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"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ] <span class="kw">means </span>$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ] <span class="kw">means </span>$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</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">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<a NAME="E6:2"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
 for <font color="Olive" title="b1">q</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">q</font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">q</font>] )
 
;<br/>
<a NAME="E7:2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">q</font>] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">r</font>] </span> } </span> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="hilb10_3.html#S2" onmouseover="rs('hilb10_3/S2')" onmouseout="rh()">HILB10_3:sch 2</a>(<a class="txt" href="#E6:2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</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"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2"><span class="lab"><font color="Green" title="E7">A8</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/>


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