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<div class="add">

<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><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> ;<br/>





<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E52">A1</font></i>: 
<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>
 ;<br/>

<a NAME="E2:73"/><b>then </b>
(  <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> &amp; ( 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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</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> ) )
 
<b>by </b><i><a class="ref" href="int_4.html#T49" target="_self" title="INT_4:th.49">Th49</a></i>;<br/>
<a NAME="E3:73"/><b>then </b><i><font color="Green" title="E53">A2</font></i>: 
 <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#K6" title="MEASURE6:func.6">++</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>
 
<b>by </b><i><a class="ref" href="int_4.html#T2" target="_self" title="INT_4:th.2">Th2</a></i>;<br/>
<a NAME="E4:73"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</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> 
 
<b>by </b><i><a class="txt" href="int_4.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<a NAME="E5:73"/>
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#K6" title="MEASURE6:func.6">++</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#K6" title="MEASURE6:func.6">++</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="73_1"><b>proof </b></a><div class="add">

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E55">A4</font></i>: 
( <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#K6" title="MEASURE6:func.6">++</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#K6" title="MEASURE6:func.6">++</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> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">u</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E56">A5</font></i>: 
( <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> &amp; <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="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">u</font> )
 <b>by </b><i><a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">w</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:73_1"/><i><font color="Green" title="E57">A6</font></i>: 
( <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> &amp; <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="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">w</font> )
 <b>by </b><i><a class="txt" href="int_4.html#E1:73_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">u'</font> = <font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c9">w'</font> = <font color="Maroon" title="c7">w</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="txt" href="int_4.html#E3:73_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="int_4.html#E5:73_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E7:73_1"/><i><font color="Green" title="E58">A7</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">u'</font>,<font color="Maroon" title="c9">w'</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>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:73"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="int_4.html#E1:73_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="int_4.html#E3:73_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="int_4.html#E5:73_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="int_4.html#T49" target="_self" title="INT_4:th.49">Th49</a></i>;<br/>
<b>assume </b><a NAME="E8:73_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>
 ;<br/>

<a NAME="E9:73_1"/><b>then </b>
<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></i>;<br/>
<a NAME="E10:73_1"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">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">u'</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">w'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E3:73_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="int_4.html#E5:73_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<a NAME="E11:73_1"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c8">u'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">w'</font>
 
;<br/>
<a NAME="E12:73_1"/><b>then </b>
<font color="Maroon" title="c8">u'</font>,<font color="Maroon" title="c9">w'</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E13:73_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_4.html#E7:73_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:73"/>
<font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</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>
 <b>by </b><i><a class="txt" href="int_4.html#E3:73"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="int_4.html#E4:73"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="int_4.html#T52" target="_self" title="INT_4:th.52">Th52</a></i>;<br/>


</div>
