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

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</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="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">y</font> &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font> implies (  not <font color="Maroon" title="c3">x</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c4">y</font> &amp;  not <font color="Maroon" title="c4">y</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c3">x</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:108"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">y</font> &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">and </span><br/><a NAME="E2:108"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c4">y</font> &amp;  not <font color="Maroon" title="c4">y</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<a NAME="E3:108"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c3">x</font> implies <font color="Maroon" title="c4">y</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="108_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:108_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c4">y</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E2:108_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="gaussint.html#T42" target="_self" onmouseover="rs('gaussint/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E3:108_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T43" target="_self" onmouseover="rs('gaussint/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E4:108_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="gaussint.html#T42" target="_self" onmouseover="rs('gaussint/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:108_1"/>
<font color="Maroon" title="c4">y</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="gaussint.html#T42" target="_self" onmouseover="rs('gaussint/T42')" onmouseout="rh()">Th42</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:108"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="108_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:108_2"/>
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:108_2"/><span class="kw">then </span>
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="gaussint.html#E70"><span class="lab"><font color="Green" title="E41">Lm10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:108_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="gaussint.html#T36" target="_self" onmouseover="rs('gaussint/T36')" onmouseout="rh()">Th36</a>, <a class="ref" href="complex1.html#T4" onmouseover="rs('complex1/T4')" onmouseout="rh()">COMPLEX1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="kw">hereby </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">  not <font color="Maroon" title="c4">y</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c3">x</font></span>
<div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">c</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E2:108_3"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c5">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 <span class="kw">and </span><br/><a NAME="E3:108_3"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="gaussint.html#T40" target="_self" onmouseover="rs('gaussint/T40')" onmouseout="rh()">Th40</a></span>;<br/><span class="kw">assume </span><a NAME="E4:108_3"/>
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">d</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E6:108_3"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font>
 ;<br/><a NAME="E7:108_3"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_3"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E6:108_3"><span class="lab"><font color="Green" title="E53">A9</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> ;<br/><span class="lab"><font color="Green" title="E55">A11</font></span>:  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c7">e</font> = 
<span class="p1">(<span class="default"><a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c5">c</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="c6">d</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 href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_3"><span class="lab"><font color="Green" title="E51">A7</font></span></a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2"><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="E1:108_3_2"/>
( <font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  or  not <font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:108_3_2_1"/>
<font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_3_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E7:108_3"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E9:108_3"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="gaussint.html#D20" target="_self" onmouseover="rs('gaussint/D20')" onmouseout="rh()">Def20</a>, <a class="ref" href="gaussint.html#T40" target="_self" onmouseover="rs('gaussint/T40')" onmouseout="rh()">Th40</a>, <a class="ref" href="gaussint.html#T41" target="_self" onmouseover="rs('gaussint/T41')" onmouseout="rh()">Th41</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:108_3_2_2"/><span class="lab"><font color="Green" title="E56">A12</font></span>: 
 not <font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:108_3_2_2"/>
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a></span>)</span> <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="c7">e</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="c3">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_3"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="gaussint.html#T34" target="_self" onmouseover="rs('gaussint/T34')" onmouseout="rh()">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E3:108_3_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108_3"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E1:108_3_2_2"><span class="lab"><font color="Green" title="E56">A12</font></span></a>, <a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="xcmplx_1.html#T7" onmouseover="rs('xcmplx_1/T7')" onmouseout="rh()">XCMPLX_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">consider </span><font color="Maroon" title="c5">c</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E7:108"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<font color="Maroon" title="c5">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 <span class="kw">and </span><br/><a NAME="E8:108"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="gaussint.html#T40" target="_self" onmouseover="rs('gaussint/T40')" onmouseout="rh()">Th40</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:108"/>
<font color="Maroon" title="c4">y</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c3">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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">d</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E11:108"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font>
 ;<br/>
<a NAME="E12:108"/><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E11:108"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c5">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> ;<br/>
<span class="lab"><font color="Green" title="E55">A17</font></span>:  <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c7">e</font> = 
<span class="p1">(<span class="default"><a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c5">c</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="c6">d</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 href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_5"><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="E1:108_5"/>
( <font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  or  not <font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_5_1"><span class="kw">suppose </span></a><a NAME="E1:108_5_1"/>
<font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E12:108"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E14:108"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="ref" href="gaussint.html#D20" target="_self" onmouseover="rs('gaussint/D20')" onmouseout="rh()">Def20</a>, <a class="ref" href="gaussint.html#T40" target="_self" onmouseover="rs('gaussint/T40')" onmouseout="rh()">Th40</a>, <a class="ref" href="gaussint.html#T41" target="_self" onmouseover="rs('gaussint/T41')" onmouseout="rh()">Th41</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_5_2"><span class="kw">suppose </span></a><a NAME="E1:108_5_2"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
 not <font color="Maroon" title="c6">d</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:108_5_2"/>
 <a href="gaussint.html#K16" title="GAUSSINT:func.16">Norm</a> <font color="Maroon" title="c3">x</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="c7">e</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="gaussint.html#K15" title="GAUSSINT:func.15">*'</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:108"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="gaussint.html#T34" target="_self" onmouseover="rs('gaussint/T34')" onmouseout="rh()">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E3:108_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:108"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E1:108_5_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="xcmplx_1.html#T7" onmouseover="rs('xcmplx_1/T7')" onmouseout="rh()">XCMPLX_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
