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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 5
 ;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E36</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:47"/><b>then </b><i><font color="Green">E37</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 3</span>)</span>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>suppose </b></a><a NAME="E1:47_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_1"/><b>then </b>
 <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:47_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 5 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_9.html#E3:47"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_9.html#E1:47"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><a NAME="E1:47_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_2"/><b>then </b>
 <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>then </b>0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 3</span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E2:47"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 3</span>)</span>

;<br/><a NAME="E4:47_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 3</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 5</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 3</span>)</span>
 ;<br/><b>hence </b><a NAME="E5:47_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_9.html#E33"><i><font color="Green">Lemma29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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