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<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM4" title="STRUCT_0:NM.4">Function</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">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> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Olive" title="b1">x</font> ) implies  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> holds <br/> ex <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">r</font> &amp; ( <font color="Olive" title="b4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:117"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Olive" title="b1">x</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> holds <br/> ex <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">r</font> &amp; ( <font color="Olive" title="b4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> implies  ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> st <br/>( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">r</font> &amp; ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:117"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</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">  ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> st <br/>( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">r</font> &amp; ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">a0</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b0</font> = <font color="Maroon" title="c5">b</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">a0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">b0</font>, <font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c7">b0</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E6:117"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T48" target="_self" onmouseover="rs('gaussint/T48')" onmouseout="rh()">Th48</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E8:117"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T48" target="_self" onmouseover="rs('gaussint/T48')" onmouseout="rh()">Th48</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">q0</font> = <font color="Maroon" title="c10">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">q0</font> = <font color="Maroon" title="c10">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">q1</font> = <font color="Maroon" title="c13">q0</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">r0</font> = <font color="Maroon" title="c6">a0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">b0</font></span>)</span> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> ;<br/>
<span class="lab"><font color="Green" title="E57">A5</font></span>:  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">q0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c13">q0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T19" onmouseover="rs('complex1/T19')" onmouseout="rh()">COMPLEX1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" onmouseover="rs('complex1/T12')" onmouseout="rh()">COMPLEX1:12</a></span>
;<br/>
<span class="lab"><font color="Green" title="E58">A6</font></span>:  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">q0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c13">q0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T19" onmouseover="rs('complex1/T19')" onmouseout="rh()">COMPLEX1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" onmouseover="rs('complex1/T12')" onmouseout="rh()">COMPLEX1:12</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">xq1</font> = <font color="Maroon" title="c8">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">q1</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> ;<br/>
<a NAME="E15:117"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/>
<a NAME="E16:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="square_1.html#T15" onmouseover="rs('square_1/T15')" onmouseout="rh()">SQUARE_1:15</a></span>;<br/>
<a NAME="E17:117"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/>
<a NAME="E18:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="square_1.html#T15" onmouseover="rs('square_1/T15')" onmouseout="rh()">SQUARE_1:15</a></span>;<br/>
<a NAME="E19:117"/>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E18:117"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E20:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A9</font></span>: 
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c16">xq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:117"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="gaussint.html#T47" target="_self" onmouseover="rs('gaussint/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">bxq1</font> = <font color="Maroon" title="c9">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">xq1</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> ;<br/>
<span class="lab"><font color="Green" title="E62">A10</font></span>: <font color="Maroon" title="c9">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">xq1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">b0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">q0</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c15">r0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
;<br/>
<a NAME="E23:117"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c17">bxq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c16">xq1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T34" target="_self" onmouseover="rs('gaussint/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E24:117"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c15">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:117"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E22:117"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E23:117"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E25:117"/>
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c7">b0</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="#E2:117"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="gaussint.html#T36" target="_self" onmouseover="rs('gaussint/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E26:117"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c9">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</a></span>;<br/>
<a NAME="E27:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A13</font></span>: 
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c15">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c9">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:117"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c13">q0</font>, <font color="Maroon" title="c19">r</font> = <font color="Maroon" title="c15">r0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T3" target="_self" onmouseover="rs('gaussint/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E29:117"/><span class="lab"><font color="Green" title="E66">A14</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">b0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E30:117"/>
<font color="Maroon" title="c6">a0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">b0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">r0</font>
 
;<br/>
<a NAME="E31:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A15</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:117"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E32:117"/><span class="lab"><font color="Green" title="E68">A16</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c15">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E53">A1</font></span></a></span>;<br/>
<a NAME="E33:117"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c7">b0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E53">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:117"/>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> st <br/>( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">r</font> &amp; ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> or <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:117"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="txt" href="#E31:117"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="txt" href="#E32:117"><span class="lab"><font color="Green" title="E68">A16</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>
