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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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">y</font>, <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">d</font>, <font color="Olive" title="b6">r</font>, <font color="Olive" title="b7">s</font>, <font color="Olive" title="b8">t</font>, <font color="Olive" title="b9">u</font>, <font color="Olive" title="b10">v</font>, <font color="Olive" title="b11">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b11">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b9">u</font>,<font color="Olive" title="b10">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b7">s</font>,<font color="Olive" title="b8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <font color="Olive" title="b10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b6">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <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="b9">u</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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b9">u</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> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b11">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b9">u</font></span>)</span> &amp; <font color="Olive" title="b8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b5">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">y</font></span>)</span> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> holds <br/>(  not <font color="Olive" title="b3">b</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b9">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">n</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font>, <font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c8">s</font>, <font color="Maroon" title="c9">t</font>, <font color="Maroon" title="c10">u</font>, <font color="Maroon" title="c11">v</font>, <font color="Maroon" title="c12">x</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <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="c10">u</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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">u</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> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">u</font></span>)</span> &amp; <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> implies (  not <font color="Maroon" title="c4">b</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n</font>) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">and </span><br/><a NAME="E4:36"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">and </span><br/><a NAME="E5:36"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:36"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <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="c10">u</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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">u</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> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:36"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">u</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:36"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:36"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</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"> (  not <font color="Maroon" title="c4">b</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n</font>) )</span><br/>

<a NAME="E10:36"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">s</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="hilb10_6.html#E5"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:36"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>) &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="hilb10_1.html#T4" onmouseover="rs('hilb10_1/T4')" onmouseout="rh()">HILB10_1:4</a></span>;<br/>
<a NAME="E13:36"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <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> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T7" onmouseover="rs('hilb10_1/T7')" onmouseout="rh()">HILB10_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:36"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>) &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="hilb10_1.html#T4" onmouseover="rs('hilb10_1/T4')" onmouseout="rh()">HILB10_1:4</a></span>;<br/>
<a NAME="E16:36"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> <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> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T7" onmouseover="rs('hilb10_1/T7')" onmouseout="rh()">HILB10_1:7</a></span>;<br/>
<a NAME="E17:36"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</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> 1
 
;<br/>
<a NAME="E18:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<font color="Maroon" title="c10">u</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">v</font> <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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a></span>;<br/>
<a NAME="E19:36"/><span class="lab"><font color="Green" title="E51">A16</font></span>: 
(  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <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="hilb10_1.html#T3" onmouseover="rs('hilb10_1/T3')" onmouseout="rh()">HILB10_1:3</a></span>;<br/>
<a NAME="E20:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A17</font></span>: 
(  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<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> 1</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> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<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> 1</span>)</span>) <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"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T6" onmouseover="rs('hilb10_1/T6')" onmouseout="rh()">HILB10_1:6</a></span>;<br/>
<a NAME="E21:36"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
( <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> <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"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> <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"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E22:36"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
( <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <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"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <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"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E23:36"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c11">v</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"> contradiction</span><br/>

<a NAME="E2:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:36"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="ref" href="nat_1.html#T15" onmouseover="rs('nat_1/T15')" onmouseout="rh()">NAT_1:15</a></span>;<br/>
<a NAME="E3:36_1"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E21:36"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E49">A14</font></span></a></span>;<br/>
<a NAME="E4:36_1"/>
<font color="Maroon" title="c4">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E5:36_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <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="#E3:36_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="xreal_1.html#T232" onmouseover="rs('xreal_1/T232')" onmouseout="rh()">XREAL_1:232</a></span>;<br/>
<a NAME="E6:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:36"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E7:36_1"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="nat_1.html#T15" onmouseover="rs('nat_1/T15')" onmouseout="rh()">NAT_1:15</a></span>;<br/>
<a NAME="E8:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="gaussint.html#T1" onmouseover="rs('gaussint/T1')" onmouseout="rh()">GAUSSINT:1</a></span>;<br/>
<a NAME="E9:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</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="txt" href="#E22:36"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="txt" href="#E13:36"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:36_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E36">A1</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="E24:36"/><span class="kw">then </span>
<font color="Maroon" title="c14">n1</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="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="hilb10_1.html#T3" onmouseover="rs('hilb10_1/T3')" onmouseout="rh()">HILB10_1:3</a></span>;<br/>
<a NAME="E25:36"/><span class="kw">then </span>
<font color="Maroon" title="c14">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E26:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="hilb10_1.html#T10" onmouseover="rs('hilb10_1/T10')" onmouseout="rh()">HILB10_1:10</a></span>;<br/>
<a NAME="E27:36"/>
<font color="Maroon" title="c10">u</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="ref" href="newton03.html#D1" onmouseover="rs('newton03/D1')" onmouseout="rh()">NEWTON03:def 1</a>, <a class="txt" href="#E26:36"><span class="lab"><font color="Green" title="E56">A24</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="E28:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A25</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">u</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">u</font> &amp; <font color="Maroon" title="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:36"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="ref" href="xreal_1.html#T97" onmouseover="rs('xreal_1/T97')" onmouseout="rh()">XREAL_1:97</a>, <a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E29:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A26</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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> <font color="Maroon" title="c1">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="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <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="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">B</font> = <font color="Maroon" title="c4">b</font> as   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E41">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E32:36"/><span class="lab"><font color="Green" title="E59">A27</font></span>: 
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>) &amp; <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="hilb10_1.html#T4" onmouseover="rs('hilb10_1/T4')" onmouseout="rh()">HILB10_1:4</a></span>;<br/>
<a NAME="E33:36"/>
<font color="Maroon" title="c7">r</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="#E5:36"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E23:36"><span class="lab"><font color="Green" title="E55">A20</font></span></a></span>;<br/>
<a NAME="E34:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A28</font></span>: 
<font color="Maroon" title="c7">r</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="E35:36"/>
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E36">A1</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="E36:36"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">y</font> <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#T66" onmouseover="rs('xreal_1/T66')" onmouseout="rh()">XREAL_1:66</a></span>;<br/>
<a NAME="E37:36"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:36"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="ref" href="xreal_1.html#T66" onmouseover="rs('xreal_1/T66')" onmouseout="rh()">XREAL_1:66</a></span>;<br/>
<a NAME="E38:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A29</font></span>: 
4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">y</font> <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#T66" onmouseover="rs('xreal_1/T66')" onmouseout="rh()">XREAL_1:66</a></span>;<br/>
<a NAME="E39:36"/>
<font color="Maroon" title="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E40:36"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u</font> <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="c10">u</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> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E41">A6</font></span></a></span>;<br/>
<a NAME="E41:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A30</font></span>: 
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c16">j</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_6.html#T16" target="_self" onmouseover="rs('hilb10_6/T16')" onmouseout="rh()">Th16</a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="int_1.html#D5" onmouseover="rs('int_1/D5')" onmouseout="rh()">INT_1:def 5</a></span>;<br/>
<a NAME="E42:36"/><span class="lab"><font color="Green" title="E63">A31</font></span>: 
<font color="Maroon" title="c13">i</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="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E19:36"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a></span>;<br/>
<a NAME="E43:36"/>
<font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u</font> <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="c5">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E42">A7</font></span></a></span>;<br/>
<a NAME="E44:36"/><span class="kw">then </span>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="ref" href="int_1.html#D5" onmouseover="rs('int_1/D5')" onmouseout="rh()">INT_1:def 5</a></span>;<br/>
<a NAME="E45:36"/><span class="kw">then </span>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c16">j</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:36"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="int_1.html#T15" onmouseover="rs('int_1/T15')" onmouseout="rh()">INT_1:15</a></span>;<br/>
<a NAME="E46:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A32</font></span>: 
( <font color="Maroon" title="c16">j</font>,<font color="Maroon" title="c13">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">n1</font> or <font color="Maroon" title="c16">j</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">n1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:36"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="txt" href="#E38:36"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="hilb10_1.html#T11" onmouseover="rs('hilb10_1/T11')" onmouseout="rh()">HILB10_1:11</a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="hilb10_6.html#T24" target="_self" onmouseover="rs('hilb10_6/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E22:36"><span class="lab"><font color="Green" title="E54">A19</font></span></a></span>;<br/>
<a NAME="E47:36"/>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">n1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="int_1.html#D3" onmouseover="rs('int_1/D3')" onmouseout="rh()">INT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">divN</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E49:36"/><span class="lab"><font color="Green" title="E65">A33</font></span>: 
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">divN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a>, <a class="ref" href="hilb10_1.html#T37" onmouseover="rs('hilb10_1/T37')" onmouseout="rh()">HILB10_1:37</a></span>;<br/>
<a NAME="E50:36"/>
4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">divN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:36"><span class="lab"><font color="Green" title="E65">A33</font></span></a></span>;<br/>
<a NAME="E51:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A34</font></span>: 
( <font color="Maroon" title="c16">j</font>,<font color="Maroon" title="c13">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> or <font color="Maroon" title="c16">j</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:36"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="int_1.html#T20" onmouseover="rs('int_1/T20')" onmouseout="rh()">INT_1:20</a></span>;<br/>
<a NAME="E52:36"/>
<font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></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> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E18:36"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E21:36"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E22:36"><span class="lab"><font color="Green" title="E54">A19</font></span></a></span>;<br/>
<a NAME="E53:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A35</font></span>: 
 <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>),<font color="Maroon" title="c16">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T24" onmouseover="rs('hilb10_1/T24')" onmouseout="rh()">HILB10_1:24</a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="int_1.html#T20" onmouseover="rs('int_1/T20')" onmouseout="rh()">INT_1:20</a></span>;<br/>
<a NAME="E54:36"/>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E43">A8</font></span></a></span>;<br/>
<a NAME="E55:36"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font>, <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">j</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T14" onmouseover="rs('int_1/T14')" onmouseout="rh()">INT_1:14</a>, <a class="ref" href="int_1.html#D5" onmouseover="rs('int_1/D5')" onmouseout="rh()">INT_1:def 5</a></span>;<br/>
<a NAME="E56:36"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c16">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:36"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="ref" href="int_1.html#T15" onmouseover="rs('int_1/T15')" onmouseout="rh()">INT_1:15</a></span>;<br/>
<a NAME="E57:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A36</font></span>: 
( <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c13">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> or <font color="Maroon" title="c3">n</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:36"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="int_1.html#T15" onmouseover="rs('int_1/T15')" onmouseout="rh()">INT_1:15</a></span>;<br/>
<a NAME="E58:36"/><span class="lab"><font color="Green" title="E69">A37</font></span>: 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</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="E59:36"/><span class="lab"><font color="Green" title="E70">A38</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E69">A37</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="E60:36"/><span class="lab"><font color="Green" title="E71">A39</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T13" onmouseover="rs('hilb10_1/T13')" onmouseout="rh()">HILB10_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">Pi</font> = <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">i</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E69">A37</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="E62:36"/><span class="lab"><font color="Green" title="E72">A40</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:36"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E69">A37</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="E63:36"/><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></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> <font color="Maroon" title="c13">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</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="#E42:36"><span class="lab"><font color="Green" title="E63">A31</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="E64:36"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">i</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="E2:36_2"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c3">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T14" onmouseover="rs('int_1/T14')" onmouseout="rh()">INT_1:14</a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E59:36"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="txt" href="#E62:36"><span class="lab"><font color="Green" title="E72">A40</font></span></a>, <a class="txt" href="hilb10_6.html#E25"><span class="lab"><font color="Green" title="E24">Lm7</font></span></a></span>;<br/>
<a NAME="E3:36_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c18">Pi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:36"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E59:36"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="txt" href="hilb10_6.html#E25"><span class="lab"><font color="Green" title="E24">Lm7</font></span></a>, <a class="ref" href="int_1.html#T21" onmouseover="rs('int_1/T21')" onmouseout="rh()">INT_1:21</a></span>;<br/>
<a NAME="E4:36_2"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T13" onmouseover="rs('hilb10_1/T13')" onmouseout="rh()">HILB10_1:13</a>, <a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/>
<a NAME="E5:36_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c13">i</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_2"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E4:36_2"><span class="lab"><font color="Green" title="E75">A43</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>
<span class="kw">hence </span><a NAME="E65:36"/>
(  not <font color="Maroon" title="c4">b</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c10">u</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:36"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E28:36"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E46">A11</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>
