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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><b>suppose </b></a><a NAME="E1:58_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E1:58"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_1.html#E2:58"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2"><b>suppose </b></a><a NAME="E1:58_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:58_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 <b>and </b><br/><a NAME="E4:58_1_2"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:58"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:58_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E2:58"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nat_1.html#E3:58_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 0
 ;<br/><a NAME="E2:58_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E3:58_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E4:58_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nat_1.html#E5:58_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E4:58_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><a NAME="E7:58_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="nat_1.html#T61" target="_self">Th61</a></i>;<br/><b>hence </b><a NAME="E8:58_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E4:58_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nat_1.html#E5:58_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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