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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">x</font>, <font color="Olive" title="b2">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">i</font> = <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">j</font> = <font color="Maroon" title="c3">y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">A</font> = <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;<br/>
<a NAME="E1:22"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
;<br/>
<a NAME="E2:22"/><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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<span class="p5"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T22" onmouseover="rs('hilb10_1/T22')" onmouseout="rh()">HILB10_1:22</a></span>;<br/>
<a NAME="E3:22"/><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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<span class="p5"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A1</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"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>),<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"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D5" onmouseover="rs('int_1/D5')" onmouseout="rh()">INT_1:def 5</a></span>;<br/>
<a NAME="E5:22"/>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T22" onmouseover="rs('hilb10_1/T22')" onmouseout="rh()">HILB10_1:22</a></span>;<br/>
<a NAME="E6:22"/><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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c3">y</font></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:22"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D5" onmouseover="rs('int_1/D5')" onmouseout="rh()">INT_1:def 5</a></span>;<br/>
<a NAME="E8:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T11" onmouseover="rs('int_4/T11')" onmouseout="rh()">INT_4:11</a></span>;<br/>
<a NAME="E9:22"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span><br/><div class="add"><a NAME="E2:22_1_1_1"/><span class="kw">then </span>
(  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D2" onmouseover="rs('absvalue/D2')" onmouseout="rh()">ABSVALUE:def 2</a>, <a class="ref" href="hilb10_1.html#T3" onmouseover="rs('hilb10_1/T3')" onmouseout="rh()">HILB10_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:22_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span><br/><div class="add"><a NAME="E2:22_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
( <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> &amp; <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</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="ref" href="absvalue.html#T18" onmouseover="rs('absvalue/T18')" onmouseout="rh()">ABSVALUE:18</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> = 
<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"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>

<br/>.= 
<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> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="absvalue.html#D2" onmouseover="rs('absvalue/D2')" onmouseout="rh()">ABSVALUE:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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> 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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>

;<br/><span class="kw">hence </span><a NAME="E4:22_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="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="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span></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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E10:22"/><span class="lab"><font color="Green" title="E25">A5</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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="E11:22"/><span class="lab"><font color="Green" title="E26">A6</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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="E12:22"/>
<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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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="E13:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</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"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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> 1</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E22">A1</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="E14:22"/>
<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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <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"><span class="p4">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p5"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E26">A6</font></span></a></span>;<br/>
<a NAME="E15:22"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p4"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</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> 1</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D5" onmouseover="rs('int_1/D5')" onmouseout="rh()">INT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22"/>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <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>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="int_1.html#T15" onmouseover="rs('int_1/T15')" onmouseout="rh()">INT_1:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
