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

<span class="kw">let </span><font color="Maroon" title="c1">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" 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> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b2">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <font color="Olive" title="b2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b1">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Olive" title="b2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" 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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <font color="Olive" title="b1">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Olive" title="b1">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font> implies <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:71"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/>

<a NAME="E2:71"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T49" target="_self" title="INT_4:th.49">Th49</a></span>;<br/>
<a NAME="E3:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T2" target="_self" title="INT_4:th.2">Th2</a></span>;<br/>
<a NAME="E4:71"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font> implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E2:71_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E5:71_1"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">and </span><br/><a NAME="E6:71_1"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="member_1.html#T143" title="MEMBER_1:th.143">MEMBER_1:143</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E8:71_1"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">w</font>
 <span class="kw">and </span><br/><a NAME="E9:71_1"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="member_1.html#T143" title="MEMBER_1:th.143">MEMBER_1:143</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">u9</font> = <font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c9">w9</font> = <font color="Maroon" title="c7">w</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E9:71_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:71_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E12:71_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>;<br/>
<a NAME="E13:71_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">u9</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">w9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E14:71_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c8">u9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">w9</font>
 
;<br/>
<a NAME="E15:71_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c8">u9</font>,<font color="Maroon" title="c9">w9</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E16:71_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">u9</font>,<font color="Maroon" title="c9">w9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="#E6:71_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E5:71_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E9:71_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="int_4.html#T49" target="_self" title="INT_4:th.49">Th49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:71_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>; <a class="txt" 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="E5:71"/>
<font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:71"/>
<font color="Maroon" title="c2">a</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="int_4.html#T52" target="_self" title="INT_4:th.52">Th52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
